tag:blogger.com,1999:blog-44908432173246997402021-07-17T02:35:37.101-07:00Math Day of the Year FactsUnknownnoreply@blogger.comBlogger12125tag:blogger.com,1999:blog-4490843217324699740.post-90727405202124995412020-11-18T16:01:00.048-08:002021-04-19T14:25:43.721-07:00Number Facts of the Year - 331 - 366<p> </p><div><div><div><span><span style="background-color: white; color: #222222;"><br /><span style="font-family: inherit;"><b>The 331st Day of the Year.</b></span></span></span></div><div>331 is a cuban prime of the first kind, [(y+1)^3 - y^3] (In this case, y = 10). Like all cuban primes of the first kind it is a centered hexagonal number. It is also a centered pentagonal number.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-CRKF50_R108/X4NFswYdYNI/AAAAAAAANNs/wt5INN0-WjACtCbWI1wcfIulywoGxadNQCLcBGAsYHQ/s300/centered%2Bhexagonal%2Bnumbers.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="245" data-original-width="300" src="https://1.bp.blogspot.com/-CRKF50_R108/X4NFswYdYNI/AAAAAAAANNs/wt5INN0-WjACtCbWI1wcfIulywoGxadNQCLcBGAsYHQ/s0/centered%2Bhexagonal%2Bnumbers.png" /></a></div><div><br /></div><div><span><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">31 is prime, and 331 is prime, and 3331 is prime and likewise for all the way up to 33333331. The next (with eight threes) is not prime, but if you permute the last two digits, you get 333333313, which is prime. Next string of threes followed by a single one is 17 threes and a one. strings of threes ending in 313, 311, and 323 seem likely to be prime. </span></span></span></div><div><span><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;"> A nice symmetric pic from Jim Wilder@wilderlab:</span><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><div class="separator" style="clear: both; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: center;"><a href="https://pbs.twimg.com/media/BaFemGMCIAELna4.png" style="color: #33aaff; margin-left: 1em; margin-right: 1em;"><img border="0" height="195" src="https://pbs.twimg.com/media/BaFemGMCIAELna4.png" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="320" /></a></div></span></span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span style="color: #222222;"><span style="background-color: white;">John Golden @mathhombre used this idea for one of his Talking Numbers Comics. Enjoy</span></span></div><div><span style="color: #222222;"><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-CE_Eb-ZbV-o/X8GCtwkwZfI/AAAAAAAANQE/OeH_m6OxsX0eFD2YhmiV4A9k1CfTrHgKwCLcBGAsYHQ/s900/Prime%2Bcomic%2B331.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="629" data-original-width="900" height="280" src="https://1.bp.blogspot.com/-CE_Eb-ZbV-o/X8GCtwkwZfI/AAAAAAAANQE/OeH_m6OxsX0eFD2YhmiV4A9k1CfTrHgKwCLcBGAsYHQ/w400-h280/Prime%2Bcomic%2B331.jpg" width="400" /></a></div><br /><span style="background-color: white;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">331 is the sum of five consecutive primes, 59, 61, 67, 71, 73.</span></span></div><div><span><span style="background-color: white; color: #222222;">It is also the sum of the first 15 semi-primes *Derek Orr</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">331 is the 67th prime and the sum of its digits is 7, a prime number with a prime order and a prime sum of its digits, and in the old days (before 1900) all its digits would have been considered prime. </span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">331 is a Happy number, it goes to one under the repeated iteration of the sum of the squares of its digits. 3^2 + 3^2 + 1 = 19, 1^2 + 9^2 = 82, 8^2 + 2^2 = 20....... eventually to one.</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222; font-family: inherit;"><span style="color: #212529; font-size: 16px;"> Alexandria, Egypt (birthplace of Euclid</span><span style="color: #212529; font-size: 16px;">) was founded in 331 BC by Alexander the Great. *Prime Curios</span></span></span></div><div><span><span style="background-color: white; color: #222222;"><span face="Roboto, sans-serif" style="color: #212529; font-size: 16px;"><br /></span></span></span></div><div><span><span style="background-color: white; color: #222222;"><span face="Roboto, sans-serif" style="color: #212529; font-size: 16px;">331 = 166^2 - 165^2</span></span></span></div><hr /><b>The 332nd Day of the Year</b><br />332= 2^2 x 83, and is the difference of two squares, 84^2 - 82^2, and the sum of three squares in more than one way, 2^2 + 2^2 + 18^2 = 6^2 + 10^2 + 14^2.</div><div><br /></div><div>The sum of the first 332 primes is prime. *Derek Orr</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">332 is the number of ways to partition 47 into non-zero triangular numbers. (</span><i style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">36 + 10 + 1 would be one such way</i><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">)</span></div><div><br /></div><div>The sum of the proper divisors of 332 is 256 = 2^8, *Derek Orr</div><div><br /></div><div>In base 3, 332 is a triple double digit number, 110022 <br /><br />332 and 333 have the same number of divisors, 6.</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">As numbers get larger and larger, it would seem that there would be fewer and fewer primes in each century of them, such as from 100 to 199. But there seem to be a large number of centuries with only six primes. There are only five year days such that between 100*n and 100*n+99 there are exactly six primes. 332 is one of them. (there are exactly six primes between 33200 and 33299)</span><br /><hr /><b>The 333rd Day of the Year</b><br />333=3^2 x 37, It is a Joy-Giver number, divisible by the sum of its digits. </div><div><br /></div><div>333 is palindrome, and the sum of its proper divisors, 161, is also a palindrome. *Derek Orr (and Derek, the sum of all its divisors, 494, is also a palindrome. ) </div><div><br /></div><div>333 is a Polignac number. It can not be formed by the sum of a power of two and a prime. (See Day 127 or 337 for Polignac's conjecture about this). </div><div> </div><div>333 in base eight is also a palindrome, 515.</div><div><br /></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">333 = 101 + 11 + 2 + 3 + 41 + 5 + 61 + 7 + 83 + 19, where each term is the smallest prime containing 0, 1, 2, ..., and 9, respectively. *Prime Curios</span></div><div><span face="Roboto, sans-serif" style="color: #212529;"><br /></span></div><div><span face="Roboto, sans-serif" style="color: #212529;">333 = 3^2 + 18^2 = 167^2 - 166^2= 57^2 - 54^2 = 23^2 - 14^2<br /></span><br />333 is also the sum of three squares in different ways, 4^2 + 11^2 + 14^2 = 8^2+ 10^2 + 13^2</div><div><br /></div><div>333 is expressible as n(4*n+1) and also as (4n+1)(4m+1)</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">There are 333 possible hexagonal polyominoes with seven cells.</span></div><div><br /></div><div><span>Not only is 3^2 + 4^2 =5^2 and,</span><br /><span>33^2 + 44^2 = 55^2 but</span><br /><span>333^2 + 444^2 = 555^2</span><br /><span>And I received this collection of additional related notes from @Expert_Says on twitter.</span><br /><br /><span>33² + 544² = 545²</span><br /><span style="font-size: small;"><br /></span><div style="font-family: inherit;"><span style="font-family: inherit; font-size: small;">333² + 55444² = 55445² </span></div><span style="font-size: small;"><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-family: inherit;"></span></span></span><br /></span><div style="font-family: inherit;"><span style="font-family: inherit; font-size: small;">3333² + 5554444² = 5554445² </span></div><span style="font-size: small;"><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-family: inherit;"></span></span></span><br /></span><div style="font-family: inherit;"><span style="font-family: inherit; font-size: small;">33333² + 555544444² = 555544445² </span></div><span style="font-size: small;"><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-family: inherit;"></span></span></span><br /></span><div style="font-family: inherit;"><span style="font-family: inherit; font-size: small;">333333² + 55555444444² = 55555444445²</span></div><br /><div style="font-family: inherit;"><span style="font-family: inherit; font-size: small;"> 3333333² + 5555554444444² = 5555554444445²</span></div><span style="font-size: small;"><span></span></span><hr /><b>The 334th Day of the Year</b><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">334 is an even semi-prime, 2 x 167, and together with 335 they form a semi-prime pair. (</span><i style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">There will be one more day pair this year that is a semi-prime pair, can you find it?</i><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">) There are only three more even semiprimes in the year.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">D. R. Kaprekar created the name "self number" for numbers that can not be made up as the sum of any number n and the sum of its digits. They are also called Colombian numbers or Devlali numbers. 1, 3, 5, 7, 9, 20, 31, are some of the smaller self-numbers, and of course, 334 is a self number.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">334 in base three looks more like a binary number, its 110101. Students might explore other numbers in base three (or four or five) that look binary (ie only made up of zeros and ones).</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">334 = 2 + 3 x 5 + 7 x 11 + 13 x 17 +19 *Derek Orr</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">334 uses the same digits in base 13 and 14, 1C9 and 19C *Derek Orr </span></div><div><br /></div><div>1^2 + 3^2 + 18^2 = 3^2 + 6^2 + 17^2 = 3^2 + 10^2 + 15^2 = 334</div><div><br /></div><div>334^4 + 1 is prime, it is the 48th such number of the year. A year has 51 such numbers .<br /><hr /><b>The 335th Day of the Year</b></div><div><span face="sans-serif" style="background-color: white; color: #202122;">335 = 5 × 67, </span></div><div><span face="sans-serif" style="color: #202122;"><br /></span>335 is the sum of four, but no fewer, squares. It is the 54th year day for which this is true. </div><div><br /></div><div>335 is the sum of all the digits from 1 to 38. </div><div><br /></div><div>335^3 is an eight digit number that has all odd digits . *Derek Orr</div><div><br /></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">2</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">335</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> is the smallest power of two which equals the sum of four consecutive primes. *Prime Curios This seems astounding to me, that such a huge number would be the first. </span>(Big number, how big is the smallest power of two that equals two (too easy, three?) consecutive primes?)<br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Lagrange's theorem tells us that each positive integer can be written as a sum of four squares (perhaps including zero), but many can be written as the sum of only one or two non-zero squares. 335 is one of the numbers that can not be written with less than four non-zero squares. The smallest examples are 7, 15, and 23. If you take any number in this sequence, and raise it to an odd positive power, you get another number in the sequence, so now </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">you </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">know that 7</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> = 343 is also not expressible as the sum of less than four non-zero squares.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><div>*Prime Curios </div><div><br /></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">There are 67 primes smaller than 335, and so 335 is divisible by the number of primes less than itself.. How common is that for integers. </span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br /><div><br /></div><div>There are 335 primes of the form n^8+1 below 10^4. *Derek Orr</div><div><br /></div><div>335 = 168^2 - 167^2 = 36^2 - 31^2. </div><div><br /></div><div>335 = 7^3 - 2^3<br /><hr /><b>The 336th Day of the Year</b></div><div><span face="sans-serif" style="background-color: white; color: #202122;">336 = 2</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">4</sup><span face="sans-serif" style="background-color: white; color: #202122;"> × 3 × 7,</span></div><div><span face="sans-serif" style="background-color: white; color: #202122;"><br /></span></div><div><span face="sans-serif" style="background-color: white; color: #202122;">336 is a Joy-Giver number, divisible by the sum of its digits</span></div><div><span face="sans-serif" style="background-color: white; color: #202122;"><br /></span></div><div><span face="sans-serif" style="background-color: white; color: #202122;">336 is the typical number of dimples on an American </span><a href="https://en.wikipedia.org/wiki/Golf" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; text-decoration-line: none;" title="Golf">golf</a><span face="sans-serif" style="background-color: white; color: #202122;"> ball. </span></div><div><br /></div><div>336 is an untouchable number, it can not be expressed as the sum of all the proper divisors of any other number. </div><div><br /></div><div>336= 85^2 - 83^2 = 44^2 - 40^2 = 31^2 - 25^2 = 25^2 - 17^2 = 20^2 - 8^2</div><div><br /></div><div>336 is the <span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">smallest number which is the sum in two ways of two primes</span><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"> with indices that are primes, </span><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">59 + 277 = 5 + 331 = 336. *Prime Curios</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">678901234567890123456.... with 336 digits is prime (Largest known number with this property) *Derek Orr</span></div><div><br /></div><div>336^2 + 337^2 + 338^2 is prime. *Derek Orr</div><div><br /></div><div>336 = 4^2 + 8^2 + 16^2 = 2^8 + 2^ 6 + 2^4</div><div><br /></div><div>There are 336 unique ways to partition 41 into prime parts. (2 + 2 + 37 would be one such, 23 + 13 + 5 would be another.)</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">336 is the product of three consecutive integers, 6*7*8 = 336</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Yesterday I mentioned LaGrange's theorem that every number can be written as the sum of four integral squares. Some can be written as the sum of four squares in many different ways which include the use of 0</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">. The number 26 can be partitioned as the sum of four squares in 336 different ways. If that sounds too trivial, tell folks 336 is a Lipschitz integer quaternion.</span><br /><hr /><b>The 337th Day of the Year</b></div><div>337 is a permutable prime, every permutation of its digits is prime, 373 and 733 are also prime, It is the 68th prime number and the last year day which is a permutable prime. There are no three digit permutable primes with all distinct digits. </div><div><br /></div><div>337 is the last year day which has an inverse that is a maximal digit prime ( 1/n has n-1 digits). for 1/337 has 336 digits. </div><div><br /></div><div>337 = 2^8 + 9^2. </div><div><b><br /></b></div><div>337 is a star number like a Chinese checker board , (which has 121 holes) but larger (how big would the home triangles be on such a board. <b> </b></div><div><b><br /></b></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">337 is a Pythagorean prime number.(A Pythagorean prime is a prime number of the form 4n + 1. Pythagorean primes are exactly the primes that are the sum of two squares (and from this derives the name in reference to the famous Pythagorean theorem.)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">The mean of the first 337 square numbers is itself a square. This is the smallest number for which this is true.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">The famous Fibonacci area paradox shows a 13x13 square converted to an 8x21 rectangle. The areas of the two figures, 13x13 + 8x21 = 337 (this illusion works with any Fibonacci number F(n) squared and a rectangle that is F(n-1) by F(n+1) ) Students must be aware that 13 x 13 = 169 is NOT equal to 8 x 21 = 168, so where is the flaw. Here is a post for a little history of these </span><a href="http://pballew.blogspot.com/2014/12/geometric-vanishes-little-history.html" style="background-color: white; color: #888888; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-decoration-line: none;" target="_blank">geometric vanishes</a><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: center;"></div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><a href="https://1.bp.blogspot.com/-YnQuhva6mtw/VlnyLzAc8sI/AAAAAAAAHOU/qO4-9rBorXc/s1600/fib_paradox.jpg" style="background-color: white; color: #888888; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" src="https://1.bp.blogspot.com/-YnQuhva6mtw/VlnyLzAc8sI/AAAAAAAAHOU/qO4-9rBorXc/s400/fib_paradox.jpg" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" /></a><br /><br />337 = 9^2 + 16^2, and 337^2 = 175^2 + 288^2</div><div>337 = 4^4 + 3^4</div><div><br /></div><div>Jim Wiler at Wilderlab pointed out that 337 = 3 x 3^3 + 4 x 4^3, the sum of seven third powers.</div><div><br /></div><div>337 and 373 are both prime, and both concatenations of the factors of 101, 3 x 37 and 37 x 3 *Prime Cuios</div><div><hr /></div></div><b>The 338th Day of the Year</b><div>338 = 2 x 13^2, 338 is the last year day that is twice a square. </div><div><br /></div><div>338 is the least number, and the only year date, for which the sum of its prime factors, 28, and its number of divisors, 6, are both perfect numbers. *Prime Curios </div><div><br /></div><div>338 has a remainder of two when divided by 3, 4, 6, 7, and 8. (Students might try to create numbers with a remainder of n when divided by three consecutive numbers.) </div><div><br /></div><div>338 = 4^4 + 3^4 + 1^4</div><div>338 is the sum of two squares in two different ways, 13^2 + 13^2 = 7^2 + 17^2. <br /><br />338 is a happy number, the iteration of sums of squares maps to one.</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">338 is the arithmetic mean of two triangular numbers.</span></div><div><hr /><b>The 339th Day of the Year</b></div><div>339 = 3*113, almost a palindrome prime factorization</div><div><br /></div><div>339 has a remainder of three when divided by 4, 6, 7. and 8. (compare to 338) </div><div><br /></div><div>339 is the sum of four 5th powers. 3^5 + 2^5 + 2^5 + 2^5.</div><div><br /></div><div>It is also the sum of four 4th powers, 4^4 + 3^4 + 1^4 + 1^4</div><div>And it is the sum of nine positive 4th powers if you want to seek them. <br /><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">339 is the fourth, and last, day of the year which can be expressed as the sum of the squares of three consecutive primes.</span> 7^2 + 11^22 + 13^2</div><div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">The 339th day of the year; the plane can be divided into 339 regions with 13 hyperbolae.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div>Just discovered the term <i>emirpimes</i> (semiprime reversed) for numbers like 339 and 933 which are semi-primes that are reversals of each other. 933 = 3 x 311, the prime factors are even reversals of each other.</div><div><br /></div><div>339 = 170^2 - 169^2 = 58^2 - 55^2, <br /><hr /><b>The 340th Day of the Year</b></div><div><span face="sans-serif" style="background-color: white; color: #202122;">340 = 2</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">2</sup><span face="sans-serif" style="background-color: white; color: #202122;"> × 5 × 17, Divisible by the number of primes beneath it (337 is the 68th prime)</span></div><div><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><br /></span></div><div><span face="sans-serif" style="background-color: white; color: #202122;">sum of eight consecutive primes (29 + 31 + 37 + 41 + 43 + 47 + 53 + 59), sum of ten consecutive primes (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53), *Wikipedia</span></div><div><span face="sans-serif" style="background-color: white; color: #202122;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">340 can also be written as the sum of consecutive primes in third way, 167 + 173 *Prime Curios. Only two more year days can be written as the sum of two consecutive primes.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">340! +1 is prime. There are only thirteen day numbers of the year for which n! +1 is prime, and 340 is the last of these.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Jim Wilder@wilderlab pointed out that 340 = 4</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">1</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 4</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 4</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 4</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">. Just think, tomorrow will be even a longer string of consecutive powers of four! (and 340 is the sum of five positive fifth powers)</span><br /><br />340^2 + 1 is prime, and 340^4 + 1 is prime. *Derek Orr There is only one more year day for which n^2 + 1 is prime, but two more for which n^4 + 1 is prime.</div><div><br /></div><div>340 = 4^2 + 18^2 = 12^2 + 14^2. </div><div><br /></div><div>And for the Geometry folks, 340 is the last year day that a regular n-gon of that number of sides can be constructed with (unmarked) straightedge and compass.<br /><hr /><b>The 341st Day of the Year</b></div><div>341 = 11 x 31</div><div><b><br /></b></div><div>341 is equal to the sum of the squares of the divisors of 16, 1^2 + 2^2 + 4^2 + 8^2 + 16^2</div><div><br /></div><div>There is no digit that can be inserted on both sides of the 4 such that 3x4x1 is prime.</div><div><br /></div><div>341 = (4^5 - 1)/3. Not just a curiosity. numbers in the sequence a(n) = (4^n-1)/3 will collapse to one following the Collatz conjecture after 2n+1 iterations. so 341 should end in 11 steps, 341 , 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1 (It seems some count iterations differently, OEIS says ends in 2n steps)</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">341 is the sum of seven consecutive primes, </span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">37 + 41 + 43 + 47 + 53 + 59 + 61</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">and 341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">341 is the smallest of the pseudoprimes base 2, disproving a Chinese math conjecture from around 500 BC. The conjecture was that p is prime IFF it divides 2<sup>2 - 2. </span></div><div><br /></div><div>for younger students that really means if you raise two to the 340th power, and divide by 341, you get a remainder of one.</div><div><br /></div>341 is a palindrome when written in bases 2 (101010101), 4 (11111), 8(525), 17(131), and 30 (bb). *Derek Orr (Only one more year day will be a binary palindrome) <div><br /></div><div>A pseudoprime n in a base b is a composite number n such that\(b^{n-1} \equiv 1 Mod_n \) for this example, that means that \(2^{341-1} \equiv 1 Mod_341 \) . <br /><br />Pseudoprimes are also called Poulet numbers, and Sarrus numbers. <tt style="background-color: white; color: #222222;">"Sarrus numbers" is after Frédéric Sarrus, who, in 1819, discovered that 341 is a counterexample to the "Chinese hypothesis" mentioned above. </tt><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><tt style="background-color: white; color: #222222;">"Poulet numbers" appears in Monografie Matematyczne 42 from 1932, apparently because Poulet in 1928 produced a list of these numbers *OEIS</tt><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="-apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif" style="background-color: rgba(0, 0, 0, 0.03); color: #0f1419; font-size: 15px; white-space: pre-wrap;">341=1²+4²+18² 341=1²+12²+14² 341=2²+9²+16² 341=4²+6²+17² 341=4²+10²+15² 341=6²+7²+16²</span><br />\(341 = 8^2 + 9^2 + 14^2 \)</div><div><br style="background-color: white;" />341 can be written as the sum of five consecutive powers of 4. (see 340)<div><br /></div><div>341 is also the sum of two (consecutive) positive cubes, 5^3 + 6^3 = 341 . Such numbers are called centered cube numbers. (<span face="Roboto, Helvetica, sans-serif" style="background-color: white; color: #111111;">A centered cube number is a centered figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with i points on the square faces of the ith layer.*Wikipedia)</span></div><div><br /></div><div>341 = 171^2 - 170^2 = 21^2 - 10^2<br /><div><hr /><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span><b>The 342nd Day of the Year</b></span></span></span></span></span></span></span></div><div><span style="font-family: inherit; font-size: small;">342 = 2 x 3^2 x 19, A Joy-Giver, or Harshad, number, divisible by the sum of its digits. </span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> 342 = 18 x 19, is an Oblong (or promic, pronic, or heteromecic) number, the product of two consecutive integers and thus twice a triangular number. There will be no more of them this year. It is the last such day of the year. (Promic numbers are related to the infinite nested iteration of roots, <a href="https://pballew.blogspot.com/2018/06/infinite-radical-sequences-still-he.html" target="_blank">as I discovered here</a></span><a href="http://www.google.co.uk/url?sa=t&rct=j&q=ballew%20nested%20roots&source=web&cd=1&ved=0CB8QFjAA&url=http%3A%2F%2Fpballew.net%2Fiteroots.doc&ei=4yDcToGeKcm_tgeY4L3sAQ&usg=AFQjCNEE6TqkcLIsk4we0_M1q-Wxdd7Slg&sig2=DMz8cbaOTgFyZM5lDX7baQ&cad=rja" style="background-color: white; color: #888888; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-decoration-line: none;">.</a><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">342 is also the sum of three positive cubes.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">OH MY! 342 in base 10 is </span><span face="verdana, arial, helvetica, sans-serif" style="background-color: white; color: #222222;">= 666 in base 7</span></span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;">342^2 = 116964, the concatenation of four squares, 1, 16, 9, and 64. and it is the product of 2^2 x 9^2 x 19^2 *Derek Orr</span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;">342 is the sum of three squares using 3, 3, and 18, or 2, 7, and 17, or 6, 9, and 15.</span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;">342^2 + 343^2 + 344^2 is a prime number *Derek Orr</span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;">342 is the sum of three positive cubes, 6^3 + 5^3 + 1^3</span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;">342 is the tenth term in the Tribonacci sequence starting with 1, 2, 2, 5, 9<br /><hr /><b>The 343rd Day of the Year</b></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">343 is a Friedman number (</span><span face="sans-serif" style="background-color: white; color: #222222;">named after </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Erich Friedman</span><span face="sans-serif" style="background-color: white; color: #222222;">, as of 2013</span><span face="sans-serif" style="background-color: white; color: #222222;"> an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Stetson University</span><span face="sans-serif" style="background-color: white; color: #222222;">, located in </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">DeLand, Florida *Wik)</span><span face="sans-serif" style="background-color: white; color: #222222;">,</span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> since it can be made up of arithmetical operations of its digits, (3+4)</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> = 343. There will be one more Friedman number this year; can you find it?</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Lagrange's theorem tells us that each positive integer can be written as a sum of four squares (perhaps including zero), but many can be written as the sum of only one or two non-zero squares. 335 is one of the numbers that can not be written with less than four non-zero squares. The smallest examples are 7, 15, and 23. If you take any number in this sequence, and raise it to an odd positive power, you get another number in the sequence, so now </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">you </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">know that 7</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> = 343 is also not expressible as the sum of less than four non-zero squares.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><div>*Prime Curios </div><div><br /></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Interestingly, the speed of sound in dry air at 20 °C (68 °F) is 343 m/s.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">343 is the smallest cube ending in 3. It is also the last cube of the year. As a perfect cube, it is also a perfect number of the second kind, the product of its aliquot parts is equal to the number itself. In 1879, E. Lionett defined a perfect number of the second kind as a number for which the product of the aliquot parts is equal to the number itself. So 343 is the 7th perfect number of the second kind. The only values that can be perfect numbers of the second kind are values in the form P*Q for primes P, Q, and P</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Benjamin Vitale@BenVitale noticed that </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>343</mn><mo stretchy="false">(</mo><msup><mn>10</mn><mi>n</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">)</mo></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-1-Frame" role="presentation" style="background-color: white; 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margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-3" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">343</span><span class="mo" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="msubsup" id="MathJax-Span-5" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.518em;"><span style="border: 0px; clip: rect(3.097em, 1000.95em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-6" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 1.013em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-7" style="border: 0px; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-8" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="mn" id="MathJax-Span-9" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span class="msubsup" id="MathJax-Span-10" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.824em;"><span style="border: 0px; clip: rect(3.033em, 1000.32em, 4.422em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-11" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.382em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-12" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-13" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.443em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.375em; width: 0px;"></span></span></nobr></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> is a palindrome for n greater than one, you get a palindrome... and if you divide or multiply the result by 7, you get a perfect square. Some simple algebra will allow any HS student to confirm.</span></div><div><br /></div><div><span style="font-family: inherit; font-size: small;"><span>343 is the only known example of x</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">2</sup><span face="sans-serif" style="background-color: white; color: #202122;">+x+1 = y</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">3</sup><span face="sans-serif" style="background-color: white; color: #202122;">, in this case, x=18,</span><span face="sans-serif" style="background-color: white; color: #202122;"> y=7. </span><br /><br /><span face="sans-serif" style="background-color: white; color: #202122;">343 is z</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">3</sup><span face="sans-serif" style="background-color: white; color: #202122;"> in a triplet (x,y,z) such that 3</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #202122;"> + 10</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">2</sup><span face="sans-serif" style="background-color: white; color: #202122;"> = 7</span><sup style="background-color: white; color: #202122; font-family: sans-serif; line-height: 1;">3</sup><span face="sans-serif" style="background-color: white; color: #202122;">. *Wik</span><br /><br />343 = 172^2 - 171^2 = 28^2 - 21^2</span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;">343 is not prime, in fact, no three digit palindrome with 4 in the middle is prime. Only one other digit is similarly not found as the middle digit of a three digit prime. </span></div><div><span style="font-family: inherit; font-size: small;"><br /></span></div><div><span style="font-family: inherit; font-size: small;">343 is the sum of the first five Odd Fibonacci Primes. *Prime Curios</span></div><div><br /></div><div> Mathematicians of Europe were challenge by Fermat in 1647 to "find a cube, that when increased by the sum of its aliquot parts, is a square." The cube, was \(7^3 = 343\) , and the sum of the cube and its aliquot divisors \( 7^3 + 1+ 7 + 7^2 = 400 = 20^2\) <br /></div><div><span style="font-family: inherit; font-size: small;"><hr /><b>The 344th Day of the Year</b></span></div><div><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">344 = 2^3 x 43. That means 344 has 8 divisors, and since it is divisible by eight, it is a refactorable number. </span></span></span></span></span></span></span></span><span face="Roboto, Helvetica, sans-serif" style="background-color: white; color: #444444;">First defined by </span><a h="ID=SERP,5461.1" href="https://www.bing.com/search?q=Curtis%20Cooper%20(mathematician)%20wikipedia&form=WIKIRE" style="background-color: white; border-bottom: 1px dashed rgb(204, 204, 204); color: #444444; font-family: Roboto, Helvetica, sans-serif; text-decoration-line: none; touch-action: manipulation;">Curtis Cooper</a>(Prof at U of Central Missouri and co-discoverer of the 43rd and 44th Mersenne Primes)<span face="Roboto, Helvetica, sans-serif" style="background-color: white; color: #444444;"> and Robert E. Kennedy (also at U of Central Missouri at the time of discovery), they were later named "refactorable" by Simon Colton (Prof at U of London).</span></div><div><br /></div><div><span style="font-family: inherit; font-size: small;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">344 is the sum of two positive cubes and of three positive cubes. There will only be one more day for the rest of the year that is the sum of two positive cubes.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">The sum of the squares and the sum of the cubes of the prime factors of 344 are both primes, ( </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>43</mn><mn>2</mn></msup><mo>=</mo><mn>1861</mn></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-1-Frame" role="presentation" style="background-color: white; border: 0px; color: #222222; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; white-space: nowrap;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 13.766em;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 11.43em;"><span style="border: 0px; clip: rect(1.14em, 1011.37em, 2.465em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-3" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-5" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-6" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-7" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-8" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-9" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-10" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-11" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-12" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-13" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-14" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-15" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.455em;"><span style="border: 0px; clip: rect(3.097em, 1000.95em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-16" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">43</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 1.013em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-17" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-18" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-19" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1861</span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.292em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.148em; width: 0px;"></span></span></nobr></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">and </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><msup><mn>43</mn><mn>3</mn></msup><mo>=</mo><mn>79351</mn></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-2-Frame" role="presentation" style="background-color: white; border: 0px; color: #222222; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; white-space: nowrap;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-20" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 14.334em;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 11.935em;"><span style="border: 0px; clip: rect(1.14em, 1011.87em, 2.465em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-21" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-22" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-23" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-24" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-25" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-26" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-27" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-28" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-29" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-30" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-31" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-32" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-33" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-34" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.455em;"><span style="border: 0px; clip: rect(3.097em, 1000.95em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-35" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">43</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 1.013em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-36" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-37" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-38" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">79351</span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.292em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.148em; width: 0px;"></span></span></nobr></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> ) *Prime Curios</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br />344 = 87^2 - 85^2 = 45^2 - 41^2</span></div><div><span style="font-size: small;"><br style="background-color: white;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">What does Groundhog Day (Feb 2) have to do with the 344th day of the year? (Soooo glad you asked!) If you start on New Year's day, and record the Phi function (number of days less than or equal to n and relatively prime to it). Now on Groundhog day, add them all up.... you get 344. .... Ok, an interesting historical note about what we call the Euler Phi function, Euler used the symbol Pi for it (1784) . Gauss chose the phi symbol(1801), and J J Sylvester gave it the name Totient(1879).</span></span></div><div><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></span></span></span></span></span></span></span></div><div><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span style="font-size: small;"><span br="" curios="" prime=""><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Of the 343 numbers between 344 and 2*344=688, there are 56 primes. Of the 688 numbers between 344^2 and 345^2 there are 55 primes. *Derek Orr</span></span></span></span></span></span></span></span></div><hr /><b>The 345th Day of the Year</b></div><div>345 = 3 x 5 x23, a Sphenic or wedge number. </div><div><br /></div><div>Normally the 345th day happens on Nov 12, or 1112, the fourth "see and say number" *Derek Orr</div><div>1, that's one 1, so 11. That's two 1's, so 21, and that has one 1 and one 2, so 1112. </div><div><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">345 is the average number of squirts from a cow's udder needed to yield a US gallon of milk. *Archimedes-lab.org (I have not personally verified this, so the proof is left to the reader)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">The numbers 345 and 184 form an unusual pair. Their sum is a square, the sum of their squares is a square, and the sum of their cubes is a square. </span><br /><br />Jim Wilder@wilderlab pointed out that the digits of 345 show up in two interesting equations, 3^2 + 4^2 = 5^2 and 3^3 + 4^3 + 5^3 = 6^3, </div><div><br /></div><div>3! + 4! + 5! +1 = 151, a palindromic prime. And 345 is the sum of the first six Fibonacci primes, Both from *Prime Curios</div><div><br /></div><div>345 = 2^8 + 9^2 + 2^3 *PB</div><div> </div><div>345 is the difference of two squares in several ways, 173^2 - 172^2 = 59^2 - 56^2 = 37^2 - 32^2 = 19^2 - 4^2,</div><div><br /></div><div>345 = 7^3 + 1^3 + 1^3</div><div><br /></div><div>The <a href="https://pballew.blogspot.com/2019/12/a-curious-property-of-vulgar-fractions.html" target="_blank">Farey sequence</a> using fractions with denominators less that 33 has 345 terms. </div><div><hr /><b>The 346th Day of the Year</b></div><div>346 = 2 x 173, There are only two more even semi-primes this year . </div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">346 is a Smith number. The sum of its digits equals the sum of the digits of its prime factors. 346 = 2 x 173 and 3+4+6 = 2+1+7+3. One more such number for a day this year. (Smith numbers were named by Albert Wilansky who noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith.) </span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">346 is also the fourth Franel number, the sum of the cubes of the terms in the nth row of the arithmetic triangle. </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>346</mn><mo>=</mo><msup><mn>1</mn><mn>3</mn></msup><mo>+</mo><msup><mn>4</mn><mn>3</mn></msup><mo>+</mo><msup><mn>6</mn><mn>3</mn></msup><mo>+</mo><msup><mn>4</mn><mn>3</mn></msup><mo>+</mo><msup><mn>1</mn><mn>3</mn></msup></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-1-Frame" role="presentation" style="background-color: white; border: 0px; color: #222222; direction: ltr; display: inline; float: none; font-size: 13.2px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; white-space: nowrap;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 15.281em;"><span style="border: 0px; display: inline-block; font-size: 15.84px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 12.693em;"><span style="border: 0px; clip: rect(1.14em, 1012.69em, 2.465em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-3" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">346</span><span class="mo" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="msubsup" id="MathJax-Span-5" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-6" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-7" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-8" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-9" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-10" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-11" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-12" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-13" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-14" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">6</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-15" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-16" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-17" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-18" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-19" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-20" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-21" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.95em;"><span style="border: 0px; clip: rect(3.097em, 1000.45em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-22" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.508em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-23" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.292em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.148em; width: 0px;"></span></span></nobr></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> The numbers are named for Swiss Mathematician Jérôme Franel (1859–1939).</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">346 = 11^2 + 15^2 </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">3461, 3463, 3467, and 3469 are all prime.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">346 in base nine is a palindrome 424. </span></div><div><hr /><b>The 347th Day of the Year</b></div><div>347 is the 69th prime, the smaller of a pair of twin primes, and a safe prime (also called Sophie Germain primes, meaning that 2 x 347 + 1 is also prime. <span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">There is only one more safe prime this year. It is also the smallest Friedman prime which is an emirp.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">And from Derek at @MathYearRound, "Adding 2 to any digit of 347 keeps it prime (547, 367 and 349 are prime)."</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">Derek's comment also points out that 347 is the smaller of a pair of twin primes. I just found out that, "(p, p+2) are twin primes if and only if p + 2 can be represented as the sum of two primes. Brun (1919)" (Brun showed that even if there are an infinity of prime pairs, the sum of their reciprocals converges.)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">There are 347 even digits before the 347th odd digit of π. (How often is it true that after 2n digits of π there are n even and n odd digits?)</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">347 is another Friedman number since 347 = 7^3 + 4. (see 343 for some history notes) </span></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">It is also the smallest Friedman prime which is an emirp.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">347 - (3 + 4 + 7) = 263, a prime, and 347 + (3 x 4 x 7) = 431, another prime. *Derek Orr</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span style="background-color: white;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">347 preceded by the digits 987654321 is prime. *Derek Orr</span></span></span></div><div><span style="background-color: white;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></span></div><div><span style="background-color: white;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">347 - 174^2 - 173^2 = 3^2 + 7^2 + 17^2 = 1^2 + 11^2 + 15^2 = 3^2 + 13^2 + 13^2.</span></span></span></div><div><span style="background-color: white;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></span></div><div><span style="background-color: white;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">347 is one of the numbers of Euler's incredible sequence of primes of the form n^2 + n + 41, when n = 17.</span></span></span></div><div><span style="background-color: white;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></span></div><div><span style="background-color: white;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">347 is a left trucatable prime. Taking off the leftmost digit forms another prime. 347 ---- 47 --- 7.</span></span></span></div><div><div><br /><hr /><b>The 348th Day of the Year</b></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">348 is the sum of four consecutive primes. It is the last day of the year that is of such distinction.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">348 is the smallest number whose fifth power contains exactly the same digits as another fifth power... find it.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br />348 =2^2 x 3 x 29 is called a refactorable number because it is divisible by the number of its divisors, (12). </div><div><br /></div><div>Derek Orr pointed out that 348 + 3 x 4 x 8 and 348 - 3 x 4 x 8 and 348 + 3 + 4 + 8 and 348 - 3 - 4 - 8 are all palindromes. </div><div><br /></div><div>348 = 88^2 - 86^2 = 32^2 - 26^2</div><div><hr /><b>The 349th Day of the Year</b><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">349 is a prime, the 70th, and the sum of three consecutive primes (109 + 113 + 127).</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">349 is the last day-number of the year that will be a member of a twin prime.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">349 is also the largest day-number that is a prime such that p- product of its digits and p+product of its digits are both also prime; for 349, 349 + 3*4*9 = 457 and 349 - 3*4*9 = 241.. and 349, 457 and 241 are all prime. *Ben Vitale</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="sans-serif" style="background-color: white; color: #202122;">349 was the winning number of the </span><a href="https://en.wikipedia.org/wiki/Pepsi_Number_Fever" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; text-decoration-line: none;" title="Pepsi Number Fever">Pepsi Number Fever</a><span face="sans-serif" style="background-color: white; color: #202122;"> grand prize draw on May 25, 1993, which had been printed on 800,000 bottles instead of the intended two. The resulting riots and lawsuits became known as the </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/349_incident" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; text-decoration-line: none;" title="349 incident">349 incident</a>. *Wik</div><div><br /></div><div>349 is the only prime less than a googol for which 7^p + 6 is a prime (for p a prime).*Prime Curios</div><div><br /></div><div>349 = 5^2 + 18^2, and 349^2 = 180^2 + 299^2</div><div><br /></div><div>the sum of 349^n for powers 0 through 6, is prime, *Derek Orr</div><div><br /></div><div>349 + 3 x 4 x 9 and 349 - 3 x 4 x 9 are both prime<br /><hr /><b style="font-family: inherit;">The 350th Day of the Year</b></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">350 is S(7,4), a Stirling Number of the second kind.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">350</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">+1 = 122,501 is prime. The last day of the year for which n</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 1 is prime.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Lucky Sevens, 350 = 7</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 7</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Both 350 and 351 are the product of four primes. 350 = 2x5x5x7 and 351 = 3x3x3x13. They are the third, and last pair of consecutive year days that are the product of four primes. (Don't just sit there, find the others!")</span></div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>350 is a pseudo-perfect number, it is the some of some, but not all, of its proper divisors. Any multiple of a pseudo-perfect number is also a pseudo-perfect number. If such a number is not divisible by a smaller pseudo-perfect number, it is called a primitive pseudo-perfect number. Pseudo-perfect numbers are also called semiperfect. </span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>350 is also divisible by the number of primes before it, 70. </span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>There are 350 non-square rectangles on a 6x6 grid. *Derek Orr</span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>The sum of the powers of 350 from zero to six, is prime *Derek Orr</span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>350 is a palindrome in Duodecimal (base 12) 252</span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>350 is expressible as the sum of three squares in three ways; 18^2 + 5^2 + 1^2 = 17^2 + 6^2 + 5^2 = 15^2 + 11^2 + 2^2</span></span></div><div><br /></div><div><span style="font-family: inherit;"><hr /><b>The 351st Day of the Year</b></span></div><div><span style="font-family: inherit;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">351 is the 26th triangular number (27 choose 2), and the sum of five consecutive primes. It is also an element in the Padovan Sequence</span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">, an interesting exploration for students. *Wik</span></span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">351 is the last year date that is a reversible triangular number, since 153 is also triangular. </span></div><div>These two numbers form two of the 11 multi-digit triangular numbers which are divisible by the sum of their digits, 351/9 = 39; 153/9 = 17. </div><div><br /></div><div><span style="font-family: inherit;">351 can not be written as the sum of three squares. It is the 85th year day for which that is true, there is only one more this year. It is also not the sum of two squares.</span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">351 is, however, the difference of two squares in four ways, 351 = 176^2 - 175^2 = 60^2 - 57^2 = 24^2 - 15^2 = 20^2 - 7^2. </span></div><div><span style="font-family: inherit;">and 351 is the sum of two positive cubes, 7^3 + 2^3 </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">351 is the smallest number whose sixth power has six zeros. (Is there another year day that has this quality?) *Derek Orr (he also points out that the first, second, and third powers use no digit greater than five. </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">351^2 + 2 is prime. </span></div><div><span style="font-family: inherit;">When x=351, x^2 + x +/- 1 form a pair of twin primes. *Derek Orr</span></div><div><span><hr style="font-family: inherit;" /><b style="font-family: inherit;">The 352nd Day of the Year</b><span style="font-family: inherit;"> </span><br /><span style="font-family: inherit;">352 is the last day of the year that appears in the Lazy Caterers Sequence, also called the pancake cutting sequence and the Central Polygonal numbers. The numbers describes the maximum number of pieces that a flat disc (or pancake) could be cut with n straight lines. For 26 straight cuts, that number is 352 pieces. The formula is given by P = \( \frac{n^2 + n + 1}{2}\) .</span></span><div><b style="font-family: inherit;"><br /></b></div><div><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; font-family: inherit; font-weight: bold; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-RsWgXHQ2mrU/Xi-KE_bGcZI/AAAAAAAALJQ/kFbubFLFPnkVh2ZjqTIAjRtsiBTYZkX6ACLcBGAsYHQ/s1600/pancake%2Bsequence.jpg" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="271" data-original-width="230" height="320" src="https://3.bp.blogspot.com/-RsWgXHQ2mrU/Xi-KE_bGcZI/AAAAAAAALJQ/kFbubFLFPnkVh2ZjqTIAjRtsiBTYZkX6ACLcBGAsYHQ/s320/pancake%2Bsequence.jpg" width="272" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Wik</td></tr></tbody></table><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">There are 352 ways to arrange 9 queens on a 9x9 chessboard so that none are attacking another. (</span><i style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">Gauss worked on the generalized queens problem; Students might try to find the number for small n x n boards. A general algorithm is not yet known</i><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><b style="text-align: center;"><br /><a href="https://4.bp.blogspot.com/-a7zCd_h_uk4/Xi-K-G9XLBI/AAAAAAAALJc/EbSIb7P0neE5eWn0gI5L1qEiovtbDh1NwCLcBGAsYHQ/s1600/pancake%2Bsequence%2Bcombinations.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="407" height="61" src="https://4.bp.blogspot.com/-a7zCd_h_uk4/Xi-K-G9XLBI/AAAAAAAALJc/EbSIb7P0neE5eWn0gI5L1qEiovtbDh1NwCLcBGAsYHQ/s400/pancake%2Bsequence%2Bcombinations.jpg" width="400" /></a></b></div><div><div style="text-align: center;"><b><br /></b></div><div style="text-align: center;"><b><br /></b></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">352 has all prime digits, and so does the 352nd prime, 2377.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>352 = 8^2 + 12^2 + 12^2</span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>352 = 89^2 - 87^2 = 46^2 - 42^2 = 26^2 - 18^2 = 18^2 - 3^2 <br /></span></span><br />352 = 173 + 179, the sum of two consecutive primes. </div><div><br /></div><div>The sum of the digits of 352 divides the product of the digits. How frequent is this. </div><div><hr style="font-family: inherit; font-weight: bold;" /><b>The 353rd Day of the Year</b></div><div><span style="font-size: small;">353 is the 71st prime number (note that 71 is prime as well) Only one more prime year day this year (or any year).</span></div><div><span style="font-size: small;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">353 is the last day of the year that is a palindromic prime. It is the first multi-digit palindromic prime with all prime digits.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Also, it is the smallest number whose 4th power is equal to the sum of four other 4th powers, as discovered by R. Norrie in 1911: 353</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> = 30</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 120</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 272</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 315</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">. *Wik *R. Norrie, University of St. Andrews 500th Anniversary Memorial Volume, Edinburgh, 1911.</span></span></div><div><span style="font-size: small;"><br />353 is the<span face="Roboto, sans-serif" style="background-color: white; color: #212529;"> sum of the first seventeen palindromic </span><span face="Roboto, sans-serif" style="background-color: white; color: #212529;">numbers, beginning with 0. *Prime Curios</span></span></div><div><span style="font-size: small;"><span face="Roboto, sans-serif" style="background-color: white; color: #212529;"><br /></span></span></div><div><span style="font-size: small;"><span face="Roboto, sans-serif" style="background-color: white; color: #212529;">353 is the smallest palindrome that is the sum of eleven consecutive primes, </span></span><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">(13+17+19+23+29+31+37+41+43+47+53=353). *Prime Curios</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">353 = 2^4 + 3^4 + 4^4 *Prime Curios</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">and similarly, 3^4 + 5^4 + 3^4 = 787, another palindromic prime. *Prime Curios</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">353 is the hypotenuse of a Pythagorean Right Triangle. 353^2 = 272^2 + 225^2 </span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">353 = 8^2 + 17^2 = 177^2 - 176^2</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span style="font-size: small;"><b><hr />The 354th Day of the Year</b></span></div><div><span style="font-size: small;">354 = 1^4 + 2^4 + 3^4 + 4^4</span></div><div><span style="font-size: small;"><br /></span></div><div><span style="font-size: small;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">354 is the sum of three distinct primes. (</span><i style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">It is also the solution to one version of an unsolved recreational math problem called the <a href="http://en.wikipedia.org/wiki/Postage_stamp_problem" style="color: #888888; text-decoration-line: none;" target="_blank">Postage Stamp Problem</a></i><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">, or sometimes Frobenius problem)</span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">354 = 2 x 3 x 59, a Sphenic number</span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">354 is the smallest number whose sum of the distinct prime factors is a cube, 2+3 + 59 = 64 = 4^3</span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">Of all the Primes less than 10^10, the largest difference between two consecutive primes is 354. *Derek Orr</span></span></span></div><div><span><hr style="font-size: medium;" /><b>The 355th Day of the Year</b></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">355 is the 12th Tribonacci number, Like Fibonacci but start with 1,1,1 and each new term is the sum of the previous three terms.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">355 is almost exactly </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>113</mn><mi>&#x03C0;</mi><mo>=</mo><mn>354.9999699..</mn></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-1-Frame" role="presentation" style="background-color: white; border: 0px; color: #222222; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; white-space: nowrap;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 11.114em;"><span style="border: 0px; display: inline-block; font-size: 15.84px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 9.22em;"><span style="border: 0px; clip: rect(1.329em, 1009.16em, 2.402em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-3" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">113</span><span class="mi" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-6" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">354.9999699..</span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 0.989em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.072em; width: 0px;"></span></span></nobr></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> No year day is closer to an integer multiple of pi. For that reason, it offers a really good approximation to Pi, 355/113. The Chinese often call this ratio Zu Lu after the Chinese mathematician and astrologer, Zu Chongzhi who found it in the 5th Century. </span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">355 is also the last Smith number of the year. A composite number with the sum of its digits equal to the sum of the digits of it's prime factors 3 + 5 + 5 = 5 + 7 + 1 (355 = 5 x 71) Numbers where each prime factor is used only once, are called hoax numbers, see 364.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">If you write out the binary expression of 355, and examine it as a decimal number, (101100011) it is prime. 355 is the last day of the year that is such a number.</span><br /><br />355 is the last day of the year that is an emirpimeS (semiprime spelled backward) . Its reversal, 553 is also a semiprime, 7 x 79.</span></div><div><span><br />Like 350, 355 is divisible by the number of Primes below it, 71</span></div><div><span><br /></span></div><div><span>355 = 178^2 - 177^2 = 38^2 - 33^2, </span></div><div><span><br /></span></div><div><span>355 is expressible as the sum of three squares in two distinct ways, 355 = 15^2 + 11^2 + 3^2 = 15^2 + 9^2 + 7^2, <br /><hr style="font-size: medium;" /><b>The 356th Day of the Year</b></span></div><div><span>When the iterated sum of the squares of digits of a number produce 1, it is called a Happy Number. If any number is Happy, and permutation of it's digits is also Happy, and inserting any number of zeros also will result in a Happy Number (13 for instance is Happy, since 1^2 + 2^2= 10, and 1^2 + 0^2= 1, so 31, 103, 301 and 310 are also Happy Numbers.<br /> 356 is happy because it follows the chain 70---49---97---130---10---1<br />I have proposed the use of the term Principle Happy numbers for those that do not contain a zero, or any reordering of a previous happy number. Here is a list of the smallest 30 Principle Happy Numbers, and makes searches more direct since no descending sequences of digits can exist. 1, 7, 13, 19, 23, 28,44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899, Note that the last day year which is a principal happy number is Day 356. (365 is the last year day that is Happy)<br /><br /></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">There are 356 ways to partition the number 36 into distinct parts without a unit.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">356 is the last day of the year that will be a self-number, (there is no number n such that n+ digit sum of n = 356)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">356 = 2</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> x 89. Numbers that are the product of a prime and the square of a prime are sometimes called Einstein numbers, after E = m c</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;"><br /></span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;">356 = 10^2 + 16^2; and 356^2 = 320^2 + 156^2.</span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;"><br /></span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;">356 = 90^2 - 88^2</span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;"><br /></span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;">356 is the sum of three squares in three unique ways. 356 = 18^2 + 4^2 + 4^2 = 16^2 + 8^2 + 6^2 = 14^2 + 12^2 + 4^2</span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;"><br /></span></span></span></div><div><span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;">The 356th Prime is 2393. Note that the concatenation of the two 3562393 is also prime. Their sum, 356 + 2393 = 2749 is prime as well. And 3 + 5 + 6 + 2 + 3 + 9 + 3 = 31.... Yep. *Prime Curios</span></span></span></div><div><span><hr style="font-size: medium;" /><b>The 357th Day of the Year</b><br />357 is the first three digits of the longest left truncatable prime now known, <span style="background-color: white; font-family: monospace; text-align: -webkit-right;">357686312646216567629137. If you mark off the leading digits one by one, after each one you still have a prime number, finishing in the prime sequence 9137, 137, 37, 7 </span><br /><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">There are 357 odd numbers in the first 46 rows of Pascal's Arithmetic triangle. (How many evens?)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">357 is made up of three consecutive prime digits, and is the product of three distinct primes, 3 x 7 x 17=357, thus a Sphenic (wedge) number.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">There are 21 year dates for which the sum of the divisors is a square number. 357 is the 20th of them. 1+3+7+17+21+51+119+357=576=24</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><br /><br />357 is made up of three distinct prime digits, yet it is not a prime number, and none of the five other permutations of the digits is prime. *Prime Curios (Wondering, without checking yet [yeah, I'm lazy] how many three digit numbers with distinct prime digits meet either, or both, of these conditions. Even 253, which has only two permutations that might be prime, has one that is, 523<br /><br /><br /><hr style="font-size: medium;" /></span></div></div><b>The 358th Day of the Year</b><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">358 is twice a prime, and the sum of six consecutive primes,</span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> </span><span face="sans-serif" style="background-color: white; color: #202122;">47 + 53 + 59 + 61 + 67 + 71</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">The sum of the first 358 prime numbers is itself a prime number.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">and in case you were curious, the 358th digit of pi (after the decimal point) is 3.</span></div><div><br /></div><div>358 is smallest number whose first two digits are distinct <i>odd</i> primes, and the third digit is their sum. *Prime Curios</div><div> So I suspect that there is a number with two distinct primes and the third digit is their sum, but one of the primes is two. Can you confirm, or deny?</div><div><br /></div><div>35+ 8 + 3 + 58 + 3 + 5 + 8 = 3 x 5 x 8 = 120</div><div><br /></div><div>Derek Orr pointed out that 358 = 2 x 179, and 2+ 179 = 181 is prime, and 2 + 1 + 7 + 9 = 19 is also prime. </div><div><br /></div><div>358 = 18^2 + 5^2 + 3^2 = 14^2 + 9^2 + 9^2</div><div><hr /><b>The 359th Day of the Year</b><div>359 is the 72nd prime of the year, and the last prime year day.<br /><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">359 is a Sophie Germain prime. If you start with n=89 and iterate 2n+1 you will get a string of primes that includes 359. (How many in all?)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">It is also the smallest Sophie Germain prime whose reversal, 953 is also a Sophie Germain prime (so 359 is a Sophie Germain Emirp, 953 x 2 + 1 gives another Sophie Germain prime, and it is an Emirp as well, unfortunately not with its 2n+1 pair.)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> (On most years this year day occurs on Christm</span><span style="background-color: white; color: #222222;">as Day, a fitting day for the last prime day of the year .)</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>e raised to the exponent of (Pi x sqrt(349)) is a 26 digit number that is less than one one-hundredth from being an integer.</span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>The three digit number beginning at digit 359 of Pi, and centered at digit 360, is 360 *Prime Curios.</span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>There are only 88 year days which are not the sum of three non-zero squares. 359 is the last one. </span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>359 is prime, and placing a three between, in front of, or behind, all produce primes. 3359, 3539 and 3593. *Derek Orr</span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>Like all odd numbers, 359 is the difference of two consecutive squares. 359 = 180^2 - 179^2 </span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span>359^33 is an 85 digit number whose digit sum is 359. It is the largest year day I have found for which n^k = digit sum of n. </span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span><hr /><b>The 360th Day of the Year</b><br />The three digit number centered at digit 360 of Pi, is 360. <span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">However 360 does occur once earlier centered at position 286.]</span></div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> Bryant Tuckerman found the Mersenne prime M19937 (which has 6000 digits) using an IBM360. *Prime Curios</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">360 is also the number of degrees in a full circle, and there is a (rather new) word for two angles that sum to 360 degrees. They are called "</span><a href="http://www.pballew.net/arithme6.html#explemen" style="background-color: white; color: #888888; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-decoration-line: none;" target="_blank">explementary</a><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">" .</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">360 is a highly composite number, it has 24 divisors, more than any other number of the year, in fact any number that is below twice its size.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">It is the smallest number that is divisible by nine of the ten numbers 1-10 (not divisible by 7) What is next, students?</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">There are 360 possible rook moves on a 6x6 chess board.*Derek Orr</span></div><div><br /></div><div>360 = 6^2 + 18^2 , and 360 ^2 = 288^2 + 216^2</div><div><br /></div><div>360 is divisible by 72, the number of primes below it.</div><div><br /></div><div>A 360 sided regular polygon is the smallest regular polygon whose angles (in degrees) are prime. *Prime Curios</div><div><br /></div><div>360 is also a refactorable or tau number, divisible by the number of its divisors. </div><div><br /></div><div><hr /><b>The 361st Day of the Year</b></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> 2</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">361 </sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">is an apocalyptic number, it contains 666. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">2</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">361</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">=4697085165547</span><b style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">666</b><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">455778961193578674054751365097816639741414581943064418050229216886927397996769537406063869952 That's 109 digits, </span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">and as 361 is the last year day that is a perfect square, important to point out for students that all perfect squares are also the sum of consecutive triangular numbers, 361= 171 + 190</span></div><div><br /></div><div>There are 361 positions on a Go Board<div><div><span style="font-family: inherit; font-size: small;"><br />361 is only the second square year date that requires the sum of five powers of two to achieve, the first since 121. 361= 2^8 + 2^6 + 2^5 + 2^3 + 2^0 </span></div><div><br /></div>One of Ramanujan's approximations of pi involved 361, pi is approximately (9 ^2 + 361/22)^ (1/4) = 3.1415926525826461252060371796440223715578779831601261496951353279 *Prime Curios </div><div><br /></div><div>Derek Orr pointed out that 361 is a square that is the concatenation of two squares, 36 and 1.</div><div><br /></div><div>361 = 181^2 - 180^2</div><div><br /></div><div>361 is the sum of three non-zero squares in three ways. Find them.</div><div><br /></div><div>361= 19^2 is the largest square year day and is expressible as three consecutive triangular numbers, 361 = 105 + 120 + 136. (Student note, All squares are the sum of two consecutive triangular numbers.)</div><div><br /></div><div>8 x 45 + 1 = 361 = 19^2. So what? Well 45 is a triangular number, and students should be aware that if you multiply any triangular number by 8 and then add one you get a square number. </div><div><br /></div><div>There are only six year days which are square and not the difference of two primes, 361 is the last one. All six of the year days with this feature are the square of primes, but for some larger numbers this is not true, both 625 and 1225 are both such squares. </div><div><br /></div><div>There are 361 terms in the Farey sequence of fractions with denominators less than or equal to 34. </div><div><hr /><b>The 362nd Day of the Year</b></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">362 and its double and triple all use the same number of digits in Roman numerals.*</span><a href="http://www2.stetson.edu/~efriedma/numbers.html" style="background-color: white; color: #888888; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-decoration-line: none;" target="_blank">What's Special About This Number.</a></div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">3!+6!+2! - 1 =727 and 3!*6!*2! + 1=8641 are both prime *Prime Curios</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">362 is the sum of 3 non-zero squares in exactly 4 ways (Collect the whole set) </span><br /><br /></div></div><div>362 = 1^2 + 19^2 = 17^2 + 8^2 + 3^2 = 16^2 + 9^2 + 5^2 = 15^2 + 11^2 + 4^2</div><div><br /></div><div>Both 3! + 6! + 2! + 1 and 3! + 6! + 2! - 1 are primes *Prime Curios</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><hr /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><b>The 363rd Day of the Year</b></span><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">363 is the sum of nine consecutive primes and is also the sum of 5 consecutive powers of three. It is the last palindrome of the year.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">363 is the numerator of the sum of the reciprocals of the first seven integers, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 363/140</span><br /><br />363 = 182^2 - 181^2 = 62^2 - 59^2 = 22^2 - 11^2</div><div><br /></div><div>132 plus its reversal, 231, = 363. This is only the 49th day of the year that is the sum of a integer and its reversal.</div><div><br /></div><div>363 is only the 45th number that is divisible by the square of its largest prime factor. </div><div><br /></div><div>The 363rd digit in the decimal expression of Pi is a one. It is the 37th one in the decimal expression of Pi. </div><div><br /></div><div>363 = 19^2 + 1^2 + 1^2 = 17^2 +7^2 + 5^2 = 13^2 + 13^2 + 5^2 </div><div><hr /><b>The 364th Day of the Year</b><br />364 = 2^2 x 7 x 13, and is the sum of 12 consecutive primes beginning with 11. </div><div><br /></div><div>364 = 3^5 + 3^4 + 3^3 + 3^2 + 3^1 + 3^0</div><div><br /></div><div>364 is a repunit in base 3, 111111, and a repdigit in base 9, 444. </div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">The 364th day of the year; 364 is the total number of gifts in the Twelve Days of Christmas song: 1+(2+1) + (3+2+1) ... which is a series of triangular numbers. The sum of the first n triangular numbers can be expressed as (n+2 Choose 3). The sums of the first n triangular numbers are called Tetrahedral Numbers. 364 is the last year day which is a tetrahedral number.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">If you put a standard 8x8 chessboard on each face of a cube, there would be 364*(below) squares. Futility closet included this note on such a cube: "British puzzle expert Henry Dudeney once set himself the task of devising a complete knight’s tour of a cube each of whose sides is a chessboard. He came up with this:</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: center;"><a href="https://1.bp.blogspot.com/-EXlvqq_tewU/VPXkzzVFLYI/AAAAAAAAGdE/pkhH_UUio1M/s1600/knights%2Btour%2Bon%2Ba%2Bcube.png" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" src="https://1.bp.blogspot.com/-EXlvqq_tewU/VPXkzzVFLYI/AAAAAAAAGdE/pkhH_UUio1M/s400/knights%2Btour%2Bon%2Ba%2Bcube.png" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" /></a></div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">If you cut out the figure, fold it into a cube and fasten it using the tabs provided, you’ll have a map of the knight’s path. It can start anywhere and make its way around the whole cube, visiting each of the 364 squares once and returning to its starting point. (*BTW, I've done the arithmetic on this, and that has to be 384 squares, but I didn't notice the discrepancy at first, so it's still here)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">The number of primes less than 364 = 3*6*4 (is this true for any other number?). This product is also the total of the numbers less than 364 which are relatively prime to it. 364 = 2^2 x 3 x 7 and 2 + 3 + 7=13. </span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">364 is the 20th (and last) Hoax number of the year, (the sum of its digits is equal to the sum of the digits of it's distinct prime divisors). Exactly half those 20 numbers, including this one, have a digit sum of 13. </span><span style="background-color: white; color: #222222; font-size: 13.2px;">Numbers that are equal to the sum of the digits of all prime factors with multiplicity (in this number 2 would be added twice) are called Smith Numbers. [666 is a Smith number, It's prime factors are 2 x 3 x 3 x 37 and the sum of their digits is 2 + 3 + 3 + (3 + 7) = 18 = 6 + 6 + 6. Numbers like 22, and 85 are both Smith numbers and hoax numbers. ]</span><br /><br />364 is the sum of three squares, 18^2 + 6^2 + 2^2 </div><div><br /></div><div>364 = 92^2 - 90^2 = 20^2 - 6^2</div><div><br /></div><div>The sum of the squares of the last three year days in a leap year is a prime, 364^2 + 365^2 + 366^2 is prime. </div><div><br /></div><div>The sum of the divisors of 364 is a perfect square. It is the 21 st such number of the year. 784 = 28^2. </div><div><br /></div><div>If you have 11 candles and decide to give them to four children, (that is each child gets 0 - 11 candles and total of all four is 11), there will be 364 possible ways to do it. The quick answer is 14 choose 3.</div><div><hr /><b>The 365th Day of the Year </b><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">The 365th (and usually last) day of the year; 365 is a centered square number, and thus the sum of two consecutive squares (13</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 14</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> ) and also one more than four times a triangular number.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">365 is the sum of two squares in two ways, 13</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 14</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> and 19</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 2</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> *Lord Karl Voldevive (and of course, it is the largest year day that is the sum of two squares.)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br />365 like all odd numbers, is the odd leg of a Pythagorean Triangle with a difference of one between the even leg and hypotenuse, 365^2 + 66612^2 = 66613^2</div><div><br style="background-color: white;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">There are 10 days during the year that are the sum of three consecutive squares. This is the last one (proof left to the reader ;-} .</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">365 = 10²+11²+12²; *jim wilder@wilderlab</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">365 is the smallest number that can be written as a sum of consecutive squares in more than one way (and all the numbers squared are consecutive.): 365 =10</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 11</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 12</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> =13</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 14</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> .</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">365 is a palindrome in base 2; 101101101</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">and base 8; 555</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">365 = 183^2 - 182^2 = 39^2 - 34^2 <br /></span></span> <hr /><b>The 366th Day of the Year</b> </div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">366 is the sum of four consecutive squares, 366 = 8 </span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2,</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">+ 9</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2,</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 10</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2,</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> + 11</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">366 however is not the sum of two squares. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">To check if an arbitrary number is the sum of two squares, factor it. If any factor p^a + 1 is divisible by four, then it is not a sum of two positive squares. For 366, the factor of three is the killer.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">366 = 2 x 3 x 61 a Sphenic, or wedge number. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">366 = 13^3 - 1, which means that in base thirteen it is a repdigit, 222. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">366 is the sum of thee squares in three different ways, 19^2 + 2^2 + 1^2 = 14^2 + 13^2 + 1^2 = 14^2 + 11^2 + 7^2</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">366 is the largest year day for which there is a perfect number with that many digits. \( 2^{606} (2^{607} - 1 ) \)</span></div><div><br /></div><div><hr /></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-37130615552900767242020-10-10T13:35:00.015-07:002021-04-21T11:03:16.021-07:00Number Facts for every Year Date, 301-330<p> <b>The 301st Day of the year</b></p>301 is the sum of three consecutive primes starting at 97<br /><br />301 = 7 × 43. *Derek Orr points out that 7 = 4+3. Can you find other semi-primes where the sum of the digits of the factors are eqaul? Can you find a product of three primes all with the samed digit sum (sounds like a "work backward" method would be best for both of those.) <br /><br />301≡1Modb for every base,b, from 2 through 6 (Sixth grade version, if you divide 301 by any number 2 through 6, you get a remainder of 1)<br /><br />Any number (such as 301) with prime factors of the form x*(x+36), where x is a prime ending in 7, will end in ...01. E.g., 301 = (7*43), 901 = (17*53), 2701 = (37*73), 3901 = (47*83), etc. *Prime Curios <br /><br />301, 302, and 303 are all semi-primes<div><br /></div><div>301, like every odd number is the difference of two squares, 151^2 - 150^2 . It is also 25^2 - 18^2 (students should expand (x+7)^2 - x^2 to see why, and when this type of relation will next be useful. <br /><hr /><b>The 302nd day of the year</b><div>301, 302, and 303 are all semi-primes</div><div><br />302 = 2 x 151 <br /><br />302 is a Happy number, it only takes three iterations of the sum of the squares of the digits to get to 1<br /><br />302^2 = 91204, five distinct digits, and 302^3 = 27543608, with eight distinct digits *Derek Orr, and *PB<br /><br />The sum of divisors of 302 is 456. 302 written in base 8 is 456 *Derek Orr. Derek also points out that 302 in base nine, is 365. The number of days in a normal year (and 2020 has definitly not been normal.)<br />There are 302 ways to play the first three moves in checkers.302 is the sum of three consecutive squares 9^2 + 10^2 + 11^2</div><br /><hr /><b>The 303rd Day of the Year,</b><div><br /></div><div>301, 302, and 303 are all semi-primes</div><div><br />303 = 3 x 101, 301, 302, and 303 are all semi-primes, and all of them have two factors for which the sum of the digits is a prime, 7 and 7 for 301, 2 and 7 for 302, and 3 and 2 for 303. <br /><br />The number of primes less than 2001 is 303. Not impressed, write it the way they did in Prime Curios and you get Pi(10^3 + 10^0 + 10^3) = 303 and it looks much more impressive... "Sell the sizzle, not the steak."<br /><br />there are 303 different bipartite graphs with 8 vertices. *What's Special About This Number<br /><br />In the Gregorian calendar, 303 is the number of years that are not leap years in a period of 400 years.<hr /><b>The 304th Day of the Year</b></div><div><br />304 = 2^4 x 19. Because it has so many factors of two, it is expressible as the difference of two squares in several ways. <br />304 /4 = 76 so 77^2 - 75^2 = 304 <br />because 304/8 = 38, 40^2 - 36^2 = 304 <br />and because 304 / 16 = 19, 23^2 - 15^2 . <br /><br />There are 304 semi-primes less than 2^10, but 304 is NOT one of them. *Derek Orr <br /><br />304 is the sum of six consecutive primes (41 + 43 + 47 + 53 + 59 + 61), sum of eight consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53), *Wik <br /><br />304 is the record number of wickets taken in English cricket season by Tich Freeman in 1928, (and I do hope they got them all back!)<br /><br />Math Joke for Halloween: Why do mathematicians confuse Halloween and Christmas? Because Oct 31 = Dec 25 (31 in base 8 (Octal) is the same quantity as 25 in Decimal)</div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /> <div>304 (3! * 0! * 4!) + 1 = 43777, a prime with a prime length. *Prime Curios<br /><hr /><b>The 305th Day of the Year</b> </div><div>305 = 5 x 61, another semi-prime, along with 301, 302, 303, and now 305. Like the others, the two factors have a sum of digits that are prime, 5 and 7. </div><div><br /></div><div>305 is the smallest odd composite which is the average of two consecutive Fiboancci numbers *Number Gossip</div><div><br /></div><div>305 has two representations as a sum of two squares, 305 = 4^2 + 17^2 = 7^2 + 16^2<br /><br />and like most numbers that end in 5, 305 is expressible as the difference of two squares. 305 = 33^2 - 28^2. Algebra students should check which numbers ending in five are not the difference of squares, and how to find out the two numbers without guess and test. </div><div><br /><hr /><b>The 306th Day of the Year </b><br />306 = 2 × 32 × 17. <br /><br />306 = 9^2 + 15^2<br />306 is the sum of four consecutive primes (71 + 73 + 79 + 83)<br /><br />306 is a pronic number, the product of two consecutive natural numbes, (16 x 17 ) which is twice the 16th triangular number.<br /><br />The 306th day of the year; 306 is the sum of four consecutive primes starting with 71.<br /><br />There are 306 triangular numbers with five digits. (students, how many triangular numbers have 3 digits... Can you calculate without listing them all?)<br /><br />306 = 92 + 92 + 122; (not impressed?) you can write the same numerals with exponents and 306 =92 + 92 + 122 Really? Still not impressed, how about 306 = 82 + 112 + 112= 82 + 112 + 112.<br /><br />3*306 + 0*306 + 6*306 - 1 is prime *Prime Curios<br /><br />There are 306 primes less than 45^2.<br /><br />306 + 1 is prime<br />306^2 + 1 is prime.<br />306^8 + 1 is prime.<br />306^16 + 1 is prime.<hr /><b>The 307th Day of the Year </b><br />The 307th day of the year; 307 is the last day of the year whose square is a palindrome, 3072=94249. The next number whose square Is a palindrome is 836. <br /><br />306 is the only number less than 100,000 with a palindromic square with an even number of digits; 8362=698896 Another Ambigram Palindrome, 180 degree rotation becomes 968869<br /><br />The smallest number that is the sum of any set of primes containing all digits 0-9 : 2 + 5 + 41 + 67 + 83 + 109 = 307.<br /><br />The largest number that you can type in Excel is 9.999 * 10307. If you type in a larger number, Excel will treat it as a character string. *Prime Curios (Is this still true? Afraid I don't use Excel much anymore.)<br /><br />307 is the smallest number that is the sum of any set of primes containing all ten digits: 2 + 5 + 41 + 67 + 83 + 109 = 307. <br />,br> Smallest of the first case of seven consecutive primes that remain prime if you eliminate their first digit. <br />What??? Ok, so 307, 311, 313, 317, 331, 337, 347, are all still primes if you drop the three in front. <br /><br />307 is prime, it and its prime index (63) are both reverse triangular numbers. Are there any more examples of this property? *Prime Curios *Prime Curios (computer search anyone?)</div><div>307 = 2^4 x 3^3 - 5^3 . Is it possible to write every prime as a in the form p^a x q^b +/- r^c where p, q, and r are one each (in any order) of 2, 3, and 5? Gary Croft demonstrated all the ones under 100, </div><div><hr /><b>The 308th Day of the Year </b><br />308 is divisible by four, so 78^2-76^2 = 308 </div><div> <br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">308 is the sum of two consecutive primes.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">If 18 circles are drawn in the plane, they can separate the plane into 308 regions. Student's might try to find the maximum number of regions for smaller numbers of circles, find a pattern, write f(n).</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">308</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 308</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">0</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 308</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">8</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> is prime *Prime Curios</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><b>Derek Orr </b>noticed that 308 times the sum of its factors, 308 (2+2+7+11) = 6776, a palindrome; and if you multiply by the sum of the digits of its factors, 308(2 + 2 + 7 + 1 + 1) =4004, you get another palindrome. </span></div><div><br /></div><div>308 = 78^2 - 76^2</div><div><br /></div><div>Derek also pointed out that there are 308 primes of the form x^4 + 1 that are smaller than 10^14, (that's the answer, but who would think to ask the question?)</div><div><hr /><b>The 309th Day of the Year</b></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">309</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">5</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">= 2,817,036,000,549. It is the smallest number whose fifth power contains all the digits 0 to 9. (Students, is there a smaller number that contains all the digits for some other power?)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">309 is the fifth semi-prime between 300 and 310. It is the only one for which the sum of the digits of its prime factors are not both primes. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">if x=309, then x^2 + x + 1 is prime. *Derek Orr; how many year days make this quadratic a prime. 1, works, 2 works... now you works....(tee hee)</span></div><div><hr /><b>The 310th Day of the Year</b></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">310 = 1234 in base six. In base 2 it repeats one period, 100,110,110</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">(1!)²+(2!)²+...+(310!)² is prime. *Math Year-Round @MathYearRound</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">The four possible 3 digit permutations of 310 all use the same digits in their squares as well, 130</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> = 16900, </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">310</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> = 96100, 103</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> = 10609,and 301</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> = 90601.</span></div><div><br /></div><div>310 in base 6 is 1234. *Derek Orr</div><div><br /></div><hr /><div><b>The 311 th Day of the Year </b></div><div>311 is the only year day, and thus the smallest prime number<b> </b>which is the sum of three, five, and seven consecutive primes. (of course, go find them!)</div><div><br /></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">311 is the eleventh three-digit </span><a class="glossary" href="https://primes.utm.edu/glossary/xpage/Prime.html" style="background-color: white; border: 1px dashed rgba(0, 51, 0, 0.25); box-sizing: border-box; color: #003300; cursor: pointer; font-family: Roboto, sans-serif; font-size: 16px; padding: 0px 2px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s;" title="glossary">prime</a><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"> for which the sum of the squares of its digits is also a </span><a class="glossary" href="https://primes.utm.edu/glossary/xpage/Prime.html" style="background-color: white; border: 1px dashed rgba(0, 51, 0, 0.25); box-sizing: border-box; color: #003300; cursor: pointer; font-family: Roboto, sans-serif; font-size: 16px; padding: 0px 2px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s;" title="glossary">prime</a><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">. Note that the sum here is eleven as well. *Prime Curios</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"> 311 is the smallest number expressible as the sum of consecutive primes in four ways. </span><span face="sans-serif" style="background-color: white; color: #202122;">It can be expressed as a sum of consecutive primes in four different ways: as a sum of three consecutive primes (101 + 103 + 107), as a sum of five consecutive primes (53 + 59 + 61 + 67 + 71), as a sum of seven consecutive primes (31 + 37 + 41 + 43 + 47 + 53 + 59), and as a sum of eleven consecutive primes (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47).</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">The smallest three-digit prime that is the sum of three different three-digit primes. (101 + 103 + 107 = 311). *Prime Curios</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">311 is the 64th </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Prime" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Prime">prime</a><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">; a </span><a href="https://en.wikipedia.org/wiki/Twin_prime" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Twin prime">twin prime</a><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> with </span><a href="https://en.wikipedia.org/wiki/313_(number)" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="313 (number)">313</a><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">; *Wik</span></div><div><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><br /></span></div><div><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">311 is a </span><a href="https://en.wikipedia.org/wiki/Strictly_non-palindromic_number" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Strictly non-palindromic number">strictly non-palindromic number</a><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">, as it is not palindromic in any base between base 2 and base 309. *Wik</span></div><div><br /></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">Every permutation of 311 is prime, 31, 13, 11, 113, 131, 311, *Derek Orr</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;">The digit sum of 311 is 5, the digit sum of 311^2 is 25 *Derek Orr</span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><span style="color: black; font-size: medium;">311 is also smallest prime after 2, in a set of five consecutive primes each whose sum of digits is prime. (311, 313, 317, 331, 337)</span></span></div><div><hr /><b>The 312th Day of the Year</b></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">the number is expressed 2222 in base five. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">[It is also a Zuckerman number (a number that is divisible by the product of its digits (which is the same as the sum of its digits making it also a Harshad (joy-giver) number *PB)) Thanks to David Brooks]. 312/6 = 52</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br />312 = 2^3 x 3 x 13. Because it is divisible by a third power of two, it is expressible as the difference of two squares in two ways, 79^2 - 77^2 = 312, and 41^2 - 37^2 = 312</div><div><br /></div><div>Derek Orr points out that the prime factors of 312, and 312 itself use only the digits 1, 2, and 3.</div><div><br style="background-color: white;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">312 = 0!*5! + 1!*4! + 2!*3! + 3!*2! + 4!*1! + 5!*0! *Derek Orr</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">and 312</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> is the sum of the cubes of the integers from 14 to 25. </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>312</mn><mn>2</mn></msup><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mn>14</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>25</mn></mrow></munderover><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-1-Frame" role="presentation" style="background-color: white; border: 0px; color: #222222; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; white-space: nowrap;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 8.715em;"><span style="border: 0px; display: inline-block; font-size: 15.84px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 7.263em;"><span style="border: 0px; clip: rect(1.076em, 1007.26em, 2.718em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-3" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.96em;"><span style="border: 0px; clip: rect(3.097em, 1001.46em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">312</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 1.518em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-5" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-6" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="munderover" id="MathJax-Span-7" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.844em;"><span style="border: 0px; clip: rect(3.033em, 1001.01em, 4.422em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-8" style="border: 0px; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∑</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.349em, 1000.76em, 4.17em, -999.997em); left: 1.076em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.479em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-9" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-10" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-11" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">25</span></span></span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.286em, 1001.77em, 4.17em, -999.997em); left: 1.076em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.658em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-12" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-13" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-14" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mo" id="MathJax-Span-15" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-16" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">14</span></span></span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-17" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.193em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.013em;"><span style="border: 0px; clip: rect(3.349em, 1000.57em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-18" style="border: 0px; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.634em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-19" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-20" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-21" style="border: 0px; display: inline; font-family: MathJax_Main; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span></span></span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.67em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.451em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; clip: rect(1px, 1px, 1px, 1px); display: inline; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>312</mn><mn>2</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mn>14</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>25</mn></mrow></munderover><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup></math></span></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> *archimedes-lab.org</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">312 is between two twin primes 311 and 313. <b> Prime Curios</b> points out that the "see and say" number of 312 shares this quality, that is 131112 (one three, one one, and one two) is between a pair of twin primes as well .</span></div><div>Patrick Honaker says this is the smallest such example. </div><div><br /></div><div>312 can be written as the sum of three squares in only one way, 4^2 + 10^2 + 14^2</div><div><br /></div><div>Derek Orr also says that there are 312 ways to place 7 non-attacking queens on an 7x8 board. (Derek, nobody plays chess on a 7x8 board.)</div><div><br /></div><div>Derek also adds that in base 7 and base 11, 312 uses the same digits, 624 and 264. </div><div><br /></div><div>And in a motion of cryptic chicanery, Derek says that for bases 18 through 36, 312 uses a letter in all but one of those bases, and challenges the reader to find the unusual which is all numerical digits.</div><hr />The 313th Day of the Year<div><div>313 is prime, and 3313 is prime, 33313(7*4759) is not, but 33311 is. Going the other way, 3133 is not, but 31333 is prime, and 313333 is prime, 5 threes on the end is not, six threes on the end is. Pursue on your own, like (see day 331 for a long sting of such primes)</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> a twin prime with 311. If you draw all the diagonals of a regular dodecagon, it has 313 intersections.(I believe this is counting the 12 vertices of the dodecagon as well as 301 interior intersections.)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">bonus fact: 313 is the only 3-digit palindromic prime that is also palindromic in base 2: 100111001 * Mario Livio</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">And thanks to the folks at The Zoo of Numbers at Archimedes Lab, I now know that 313 is Donald Duck's License plate number. (But I have also seen Disney authorized materials with license DD-13)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: center;"></div><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: center;"><a href="https://2.bp.blogspot.com/-8kZ7papSdtg/WBpQexc4BcI/AAAAAAAAIkw/sEkhFlt4wLgxnpQ2sH0Ct96QpjW1BGlmACLcB/s1600/donald%2Bduck%2Bcar%2Bstamp.gif" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="320" src="https://2.bp.blogspot.com/-8kZ7papSdtg/WBpQexc4BcI/AAAAAAAAIkw/sEkhFlt4wLgxnpQ2sH0Ct96QpjW1BGlmACLcB/s320/donald%2Bduck%2Bcar%2Bstamp.gif" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="232" /></a></div></div><div><br /></div><div>313 is the sixth multidigit palindromic prime, and the last which is a year day. (11, 101, 131, 151, 181, 313,)</div><div><br /></div><div>313 is the 65th prime number, and the larger of a prime pair with 311. This is the seventh prime to have a sum of digits of seven . *Prime Curios</div><div><br /></div><div>313 is the sum of two squares, 12^2 + 13^2 , and is the hypotenuse of a Pythagorean triangle, 25^2 + 312^2 = 313^2. </div><div><br /></div><div>313 = 2^5 x 3^2 + 5^2 . Is it possible to write every prime as a in the form p^a x q^b +/- r^c where p, q, and r are one each (in any order) of 2, 3, and 5? Gary Croft demonstrated all the ones under 100, Also see 307 on this page.</div><div><br /></div><div>Frenicle challenged Wallis to solve 113 x^2 + 1 = y^2. </div><div><br /></div><div>313 is the smallest three-digit prime that is not the sum of consecutive composite numbers. *Prime Curios</div><div><br /></div><div>313 is the sum of the first 63 digits of pi. The sum of the first n digits is prime for only 11 year days. The largest or eleventh, is the sum of the first 63 digits of pi, 3 + 1 + 4 + 1 ..... </div><div><br /></div><div><br /></div><div>313 is the sum of 144, the largest Fibonacci square, and 169, the largest Pell square</div><div><hr /><b>The 314th Day of the Year</b></div><div>314 is the first three digits of Pi</div><div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">The 314th day of the year; 314 is the smallest number that can be written as the sum of of 3 positive distinct squares in 6 ways. *What's Special about this number</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">(Students, can you find the smallest number that can be written as the sum of two distinct squares in at least two ways?)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">314 is a semi-prime (2 x 157), and 314</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> +1 is prime</span><br />Another semi-prime whose factors both have a sum of digits that are both prime. <br /><br />314 = 5^2 + 17^2</div><div><br /></div><div>314 is 222 in duodecimal </div><div><br /></div><div>x^2 + x + 1 is prime if x = 314 (and some others) </div><div><br /></div><div><hr /><b>The 315th Day of the Year</b></div><div>315 = 3^2 x 5 x 7 (or 5 x 7 x 9) </div><div><br /></div><div>Since 315 is divisible by the sum of its digits, it is a Harshad, or Joy-giver number. Its also divisible by the product of its digits, or a Zuckerman number. There are only 13 three digit numbers that have both these qualities combined, and 9 of them are year days. 315 is the largest year day with this property.</div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">315</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> can be written as the sum of the cubes of five consecutive integers. Find them. (Students may also wish to find the smallest square that can be expressed as the sum of the cubes of two or more consecutive integers.)</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><span style="color: #222222;"><span style="background-color: white;">Remember those pictures you see where they ask you how many triangles in a picture like this one"<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-xpGtlVFySs0/YDgGqgqVHyI/AAAAAAAANY4/3HmpLmtsvGEc2d015E1ERajJLP6ZG9QywCLcBGAsYHQ/s298/how%2Bmany%2Btriangles.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="230" data-original-width="298" src="https://1.bp.blogspot.com/-xpGtlVFySs0/YDgGqgqVHyI/AAAAAAAANY4/3HmpLmtsvGEc2d015E1ERajJLP6ZG9QywCLcBGAsYHQ/s0/how%2Bmany%2Btriangles.jpg" /></a></div><div class="separator" style="clear: both; text-align: left;">Well the sequence of numbers depends on how many triangles on a side. Oeis gives the sequence as </div></span></span><span style="background-color: white; font-family: monospace; font-size: 13px;">1, 5, 13, 27, 48, 78, 118, 170, 235, 315 and thus with ten toothpicks on a side of the largest triangle, 315 is the largest year day which answers the problem of "How Many Triangles?" In the one shown above, search for the 16 triangles with one unit sides (10 pointed up, six down), then 7 with two unit sides (6 up one down), then 3 with three unit sides and one with four unit sides... for 27).</span></div><div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">315 is a (barely)deficient number, the sum of it's proper divisors is only 309 ...309/315 is about .9809 *</span><span class="fullname js-action-profile-name show-popup-with-id" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Derek Orr</span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> @</span><span class="username js-action-profile-name" data-aria-label-part="" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Derektionary<b> </b>pointed out to me that 256 is the closet to one of any (non-perfect) year day, (255/256 = .996), and for non-powers of two, 136 with a ratio of 134/136 = .985 is the best.</span></div><div><br /></div><div>Derek Orr pointed out that 315 times its reversal, 513 forms a six digit number that is the concatenation of two palindromes, 161595</div><div><br /></div><div>On a non leap year, the 315th day is 11/11 </div><div><br /></div><div>315 is the difference of two squares in more than one way. 158^2 - 157^2 and 26^2 - 19^2 and 54^2 - 51^2, The first is a property of all odd numbers, For a clue to the second, see Day 301 , For the third, think of how it is similar to the second. </div><div><br /></div><div> There are only 13 three digit numbers that have this property, and 9 of them are year days. 315 is the largest year day with this property.</div><div><hr /><b>The 316th Day of the Year</b></div><div><b>316 = 2^2 x 79</b></div><div><b><br /></b></div><div>316 = 7^3 - 3^3 = 80^2 - 78^2 </div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">316 can be written as the sum of 3 consecutive triangular numbers.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">316 is also a centered triangular number (these two definitions turn out to be identical, all sums of 3 consecutive triangular numbers are centered triangular numbers, and vice-versa) The nth Centered triangle number has a single point at the center (the first centered triangular number) , and triangles surrounding it with 2,3,..., n dots on a side. The fourth centered triangular, 19 =3+6+10, number is shown</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: center;"><a href="https://2.bp.blogspot.com/-ZeIvh7AD9jU/VjvAKxX-4aI/AAAAAAAAHIs/uW46HUVhz4M/s1600/centered%2Btriangle%2Bnumbers.png" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" src="https://2.bp.blogspot.com/-ZeIvh7AD9jU/VjvAKxX-4aI/AAAAAAAAHIs/uW46HUVhz4M/s320/centered%2Btriangle%2Bnumbers.png" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" /></a></div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><b></b></span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Students often enjoy finding cycles in the "sort then add" sequence. 316 is the only year date that does not reach a sorted number to terminate(at least we haven't found it yet) . [If a number is not sorted, then add n to sorted n, for example 21 is not sorted, so f(21)= 21+12=33 (a sorted number) 65 is not sorted, so f(65)= 65+56 = 121, still not sorted so do 121+112= 233, a sorted number so f(65) terminates in 233]</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">Derek Orr pointed out that there are 316 primes of the form x^2 -1 below 10,000,000</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">Derek also pointed out that 316 in base 7 uses the same digits, 631.<br /></span><hr /><b>The 317th Day of the Year</b> </div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">317 is the 66th prime number. The number made up of 317 consecutive ones digits is also prime. It is the fourth prime repunit. The smallest is 11. Find the other two. *Wik John Carlos Baez points out that, "That's not a coincidence. A number whose digits are all 1 can only be prime if the number of digits is prime! This works in any base, not just base ten. Can you see the quick proof?"</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br />317 is the only prime year date, p, such that 2^p + p is prime. The prime is 96 digits long.</div><div> <br />Derek Orr pointed out that that number with 317 ones, is (10^317 -1)/9 </div><div><br style="background-color: white;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">Not only is 317 the sum of two squares, (11^2 + 14^2) but 317</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"> is also the sum of two squares (75^2 + 308^2, making 317, 308, 75 a Pythagorean Right triangle. Can you find other primes for which both these conditions are true?</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">317 is the largest year day which is a Russian Doll Prime (Right Truncatable Prime). A number which remains a prime as each rightmost digit is stripped away, 317, 31, 3. There are only a total of 83 such numbers, the largest of which is the eight digit 73,939,133.</span></div><div><br /></div><div>If you square the digits of 317, you get 59, together you have all the odd digits. *Prime Curios</div><div><br /></div><div>\(317 = (-3)^3 +1^3 + 7^3 \) *Prime Curios</div><div><br /></div><div>Derek Orr recognized that 317 is the concatenation of two primes in two different ways, 31, 7 and 3,17. </div><div><br /></div><div>317 has a remainder of two when divided by 3, 5, 7 or 9</div><div><hr style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><b>The 318th Day of the Year</b></div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">According to Police chief Wiggum on The Simpsons; 318 is the Police code for waking a police officer in episode 5F06 Reality Bites.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">If 22 is partitioned into distinct integer parts in all possible ways, there will be 318 total parts.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">318 is a palindrome in base 9, 383</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">9</sub></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><span style="font-size: 13.3333px;"><br /></span></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">318 = 2 x 3 x 53, a sphenic number (from the Greek for "wedge")</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">318 in base three is a double digit palindrome, 10 22 10</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">318 has a remainder of three when divided by five, seven, or nine</span></div><div><hr /><b>The 319th Day of the Year</b></div><div>319 = 11 x 29, another semi-prime with the sum of the digits of its factors are both prime, 2 and 11. </div><div>It is the seventh semi-prime since 300, and six of them have the sum of the digits of their factors all prime. </div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">319 is the sum of three consecutive primes (find them). </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">319 also is the largest number whose cube has all distinct digits 319</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">3</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> =32461759. What is the largest square with all distinct digits?</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">319 cannot be represented as the sum of fewer than 19 fourth powers: 319 = 3 x 3</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 4 x 2</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> + 12 x 1</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">4</sup><br /><br />There are six ways to concatenate the number 319 and its factors, 11 and 29. Each of them is a semi-prime as well. *Prime Curios</div><div><br /></div><div>The sum of the digits of 319 is 13, the sum of the digits of its prime factors, 11 and 29, is also 13. A Smith Number.</div><div><br /></div><div>And 319 is a Happy number. The iteration of the sum of the squares of its digits leads to one, \(3^2 + 1^2 + 9^2 = 91\); \( 9^2 + 1^2= 82\); \(8^2 + 2^2 = 68\); \(6^2 + 8^2 = 100\), and \(1^2 + 0^2 + 0^2 = 1\).</div><div><br /></div><div> 319 is another that has the same remainder when divided by 5, 7 or 9, each having a remainder of 4.</div><div><br /></div><div>Derek Orr added that the first prime after 10^29 is 10^29 + 319 </div><div><br /></div><div><hr />The 320th Day of the Year</div><div>320 is the ninth Leyland Number. A number of the form x^y + y^x where both x and y are greater than one. \(8^2 + 2^8 = 320\). Named for Paul Leyland, a British number theorist.</div><div><br /></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> 320 is the maximum value of the determinant of a 10x10 binary matrix (all entries are either one or zero). (Students might explore all possible determinants of smaller matrices looking for a pattern)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">320!+1 is prime.</span><br /><br />4^2 + 8^2 = 80, so 8^2 + 16^2 = 320. </div><div><br /></div><div>320 = 2^6 x 5. With so many factors of two, it has to have several expressions as the difference of two squares, \(81^2 - 79^2 = 42^2 - 38^2 = 24^2 - 16^2 = 18^2 - 2^2 = 320\)</div><div><br /></div><div>320! + 1 is prime, *Derek Orr</div><div><hr /><b>The 321st Day of the Year</b></div><div>321 = 3 * 107</div><div><b><br /></b></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">321 is the number of partitions of 13 into at most 4 parts. (7+3+2+1 would be one such)</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">321 is a Central Delannoy number. The Delannoy numbers are the number of lattice paths from (0, 0) to (b, a) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed. Central Delannoy numbers are paths to (a,a). [Delannoy numbers are named for Henri Auguste Delannoy (1833–1915) who was a friend and correspondent of Edouard Lucas, editor of Récréations Mathématiques.]</span><br /><br />\(e ^ (\Pi * \sqrt{321})\) = 2784914870820244444545897.963...... almost an integer</div><div><br /></div><div>3-2-1, and the sum of the divisors is 4-3-2. *Derek Orr</div><div><br /></div><div>3! -2! +1! and 3! -2! -1! are both prime </div><div><br /></div><div>Using the digits of 321 exactly once each, it is possible to express two different numbers as the sum of primes, 2 + 13 = 15; and 2+31 = 33. (You can, of course, do the same with all permutations of 1, 2, 3. </div><div><br /></div><div>567^2 = 321489 using all nine non-zero digits. @fermatslibrary</div><div><br /></div><div>Derek Orr pointed out that the digit 8 appears 321 times in all the four digit primes (where does he get these things??)</div><div><hr /><b>The 322nd Day of the Year</b></div><div>322 = 2 x 7 x 23 another sphenic or "wedge" number <span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">which means that it is the area of a rectangular box (parallelepiped) with prime lengths for its length, width, and height. The sphenoid bone in the skull is so named because it is essentially a box like shape, but has two hollow sinus cavities.</span></div><div><br /></div><div>322 is a Harshad (Joy-Giver ) number divisible by the sum of its digits.</div><div><b><br /></b><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">322 is the 12th Lucas Number. The Lucas Sequence is similar to the Fibonacci sequence with L(1) = 1 and L(2) = 3 and each term is the sum of the two previous terms. L(n) is also the integer nearest to </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03D5;</mi><mi>n</mi></msup></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-1-Frame" role="presentation" style="background-color: white; border: 0px; color: #222222; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; white-space: nowrap;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 1.329em;"><span style="border: 0px; display: inline-block; font-size: 15.84px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.076em;"><span style="border: 0px; clip: rect(1.329em, 1001.08em, 2.592em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-3" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.076em;"><span style="border: 0px; clip: rect(3.097em, 1000.57em, 4.359em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ϕ</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.571em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-5" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 11.1989px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.216em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.299em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; clip: rect(1px, 1px, 1px, 1px); display: inline; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ϕ</mi><mi>n</mi></msup></math></span></span><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"> This is the last day of the year that will be a Lucas Number.</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">322 is smallest number whose square has 6 diff digits (103684). *Derek Orr</span></div><div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" />\(3^n + 2^n + 2^n\) is prime for n = 0, 1, 2, 3, 4, 5, and 6. *Prime Curios (surely you wonder if there is another power greater than six for which this is prime... SURELY!)</div><div><br /></div><div>321, 322, and 323 form the sides of an almost equilateral triangle. </div><div><br /></div><div>322 can be written as the sum of three squares in two different ways. </div><div>\(3^2 + 12^2 + 13^2 = 4^2 + 9^2 + 15^2 = 322\)</div><div><br /></div><div><hr /><b>The 323rd Day of the Year</b> <br />323 is a palindromic semi-prime with all prime digits that is the product of a twin prime pair, 17 x 19 Prime Curios says it is the smallest such palindrome. No other such palindrome is known. </div><div><br /></div><div><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">If you drew every possible path from (0,0) to (8,0) that never dropped below the x-axis using only unit vectorial moves with slopes of 1, 0, or -1 there are 323 possible paths. (alternatively this is the number of different ways of drawing non-intersecting chords on a circle with eight points- this is deceptive because it counts each way of drawing a single chord, and drawing no chords at all, students might want to count how many ways this can be done using four chords.) These are called Motzkin numbers, after Theodore Motzkin.</span></span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /> This is the eighth such number.</div><div><br style="background-color: white;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">323 is the sum of nine consecutive primes 323 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 *Derek Orr</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" />It is also the sum of 13 consecutive primes, <span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">(5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47) *Wik</span></div><div><span style="font-family: inherit;"><span style="color: #202122;"><br style="background-color: white;" /></span><span style="background-color: white; color: #222222;">323 is a palindrome and also the smallest composite number n that divides the (n+1)st Fibonacci number. *What's Special about this Number</span><br style="background-color: white; color: #222222;" /></span><br />The concatenation of 323 with itself, 323323 = 7 x 11 x 13 x 17 x 19, five consecutive primes. The sum of the squares of those primes is 989, another palindrome. *Prime Curios</div><div><br /></div><div>The sum of the digits of its prime factors (1 + 7 + 1 + 8) is the same as the product of its digits )3 x 2 x 3.) *Derek Orr</div><div><br /></div><div>Derek Orr also pointed out that inserting a 9 between any two digits of 323 forms a prime, 3923 and 3293. He didn't mention it, but 39293 is also prime. </div><div><hr /><b>The 324th Day of the Year <br /></b>324 = 2^2 x 3^4 = 4 x 3^4 = 18^2</div><div><br /></div><div><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">324 is the sum of four consecutive primes (73 + 79 + 83 + 89)*Wik</span></div><div><span face="sans-serif" style="color: #202122;"><span style="font-size: 14px;"><br /></span></span></div><div><span face="sans-serif" style="color: #202122;"><span style="font-size: 14px;">324 is divisible by the sum of its digits, and thus a Harshad or Joy-Giver number. </span></span></div><div><span face="sans-serif" style="color: #202122;"><span style="font-size: 14px;"><br /></span></span></div><div><span face="Roboto, Helvetica, sans-serif" style="background-color: white; color: #111111; font-size: 16px;">An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer, 324 is the smallest untouchable number which is a perfect square. *Prime Curios</span></div><div><span face="Roboto, Helvetica, sans-serif" style="color: #111111;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">324 is the largest possible product of positive integers with a sum of 16. (Students, Can you find the integers. Try to find the similar maximum product with a sum of 17)).</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">If you have a square array of 324 dots (that's 18x18) you can carefully paint them each in one of four colors so that no four corners of a rectangle (with sides horizontal and vertical) are the same color. you can also do that for any smaller square, but not for any larger. Here is a 17x17 to ponder</span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: center;"><a href="https://2.bp.blogspot.com/--FsJnzowj8A/VqA4tbWr_yI/AAAAAAAAHlQ/1BnzbGXNrG8/s1600/rect17.png" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="320" src="https://2.bp.blogspot.com/--FsJnzowj8A/VqA4tbWr_yI/AAAAAAAAHlQ/1BnzbGXNrG8/s320/rect17.png" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="320" /></a></div><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: center;"><br /></div><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">324 times its reversal, 423, is equal to 137052, all the non-composite digits. *Derek Orr</div><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;"><br /></div><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">324 in base 2 combines the first three powers of ten, 101000100.</div><hr /><b>The 325th Day of the Year</b></div><div>325 = 5^22 x 13, the 25th Triangular number.</div><div>(Student Note: multiply any triangular number by 9 and add 1, you get another triangular number. 9 x 36 + 1 = 325. )</div><div><br /></div><div>325 is the smallest number that can be expressed as the sum of two squares in three different ways, \(325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2 \)</div><div><br style="background-color: white;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">325 is last year day that is the sum of the first n^2 integers, </span></div><div><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>325</mn><mo>=</mo><munderover><mo movablelimits="false">&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><msup><mn>5</mn><mn>2</mn></msup></mrow></munderover><mi>i</mi></math>" face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" id="MathJax-Element-2-Frame" role="presentation" style="background-color: white; border: 0px; color: #222222; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; white-space: nowrap;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-6" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.432em;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.485em;"><span style="border: 0px; clip: rect(0.382em, 1004.42em, 3.412em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.206em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-7" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-8" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">325</span><span class="mo" id="MathJax-Span-9" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="munderover" id="MathJax-Span-10" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.256em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.14em;"><span style="border: 0px; clip: rect(3.033em, 1001.01em, 4.422em, -999.997em); left: 0.066em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-11" style="border: 0px; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∑</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.349em, 1001.08em, 4.296em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.09em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-12" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-13" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-14" style="border: 0px; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">i</span><span class="mo" id="MathJax-Span-15" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-16" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span></span></span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.097em, 1000.63em, 4.17em, -999.997em); left: 0.256em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.921em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-17" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-18" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-19" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.634em;"><span style="border: 0px; clip: rect(3.349em, 1000.32em, 4.17em, -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.974em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-20" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">5</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.382em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.227em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-21" style="border: 0px; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.98em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-22" style="border: 0px; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.193em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">i</span></span><span style="border: 0px; display: inline-block; height: 2.213em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 3.337em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -1.284em; width: 0px;"></span></span></nobr></span><br style="background-color: white; color: #222222;" />325 is a palindrome in base 2, 101000101; in base 4, 11011; and in base eight, 505. </div><div><br style="background-color: white;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">On an infinite chessboard, there are 325 different squares that can be reached in 5 knight moves.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">325 has a remainder of one when divided by 2, 3, 4, 6, 9 or 12</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">For the numbers less than 10^6, there are 325 number n for which n, n+6, n+12, and n+18 are all prime. *Prime Curios</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">325 is the hypotenuse of two Pythagorean Triangles. (36, 323, 325 and 204, 253, 325)</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">325 is the smallest triangular number that is the average of the squares of a pair of twin primes, 17 and 19. (Bet you are wondering what the next one is... huh, aren't you?..... well???)</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">3251, 3253, 3257, 3259 are all prime. *Derek Orr</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;"><br /></span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="color: #222222;">and Derek also has 325 is a palindrome when written in base 2, 4, 8, 18, and 24<br /></span><hr /><b>The 326th Day of the Year</b></div><div>326 = 2 x 163, a semiprime. Of the nine numbers from 321 to 329, six are semi-primes. </div><div><br /></div><div>326 x ( 3 x 2 x 6) = 362 a reordering of its digits. *Derek Orr</div><div><br /></div><div>An interesting combination of two statements from Derek Orr, students should consider the mutual implications:</div><div>A) Of all the the four digit primes there are 326 fours,</div><div>B) There are 326 primes less than 10,000 with at least one four. </div><div><br /></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">326 is the maximum number of pieces that may be produced in a pizza with 25 straight cuts. These are sometimes called </span><a href="http://en.wikipedia.org/wiki/Lazy_caterer%27s_sequence" style="background-color: white; color: #888888; text-decoration-line: none;" target="_blank">"lazy caterer numbers"</a><span style="background-color: white; color: #222222;"> and more generally they are centered polygonal numbers.</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">326 is also the sum of the first 14 consecutive odd primes: 326 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47. *MAA</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">326 prefixed or followed by any digit still remains composite. *Derek's Daily Math I would think such a number might be called an <i>anti-prime </i>. Wondering if there are two digit anti-primes? (seems both 20 and 32 are easy examples) </span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">How many primes can be formed with n326m (with n and m not necessarily distinct) ? I found 13263 is prime. How many are there in all. </span></span></div><div><br /></div><div>326 is a palindrome in base 12, 232.<br /><br />Unless I missed something, 326 is the smallest number missing in the Prime Curios listing. </div><div><br /></div><div>Perhaps this one from Derek Orr can capture their interest, the 54th Prime is 251, and the 54th composite is 75. SO WHAT? 251 + 75 = 326. <br /><hr /><b>The 327th Day of the Year</b></div><div>327 = 3 x 109</div><div><br /></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">327 is the largest number n so that n, 2n, and 3n together contain every digit from 1-9 exactly once. (Students might search for a smaller number with that quantity) *What's Special About This Number</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">and from Jim Wilder @wilderlab:</span><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">For day 327: 327 is a perfect totient number- φ(327)=216, φ(216)=72, φ(72)=24, φ(24)=8, φ(8)=4, φ(4)=2, φ(2)=1, and 216+72+24+8+4+2+1=327.</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">The number 327 in base ten is equal to \( 57_{[64]} \) but also \( 75_{[46]} \) </span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">327 cannot be written as the sum of three squares. Gauss found that this is only true for numbers of the form 4^k (8n-1), such as 7, 15, 23, .... but also 28, 60, 92, ... and 112, 240, etc.</span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">327 is not prime, but one in front, 1327, or a one in the back, 3271 makes it prime, but not both. </span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">There are 327 primes less than three to the seventh (3 2 7) *Derek Orr<br /><br />In the Collatz (or 3n+1) sequence, no year day takes so long as 327, which takes 143 steps to reach 1.<br /><hr /><b>The 328th Day of the Year</b></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">328 = 2^3 x 41, </span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">Another number that is three powers of ten in binary, and this one in order, 101001000. </span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif">328 is the sum of the first fifteen primes. It is the last year day that is the sum of the first n squares.</span></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif">It is also is a tau-number since it is divisible by the number of divisors it has.</span><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><br />\(328 =18^2 + 2^2 \) and</span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">\(328 = 6^2 + 6^2 + 16^2\)</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif">328 reversed is prime, but 823 can't be written as the sum of two squares, or as the sum of three squares.</span></span></span></div><div><br /></div><div><span><span style="background-color: white; color: #222222;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif">3280123456789 is prime *Derek Orr<br /></span><br />328 can be written as the difference of two squares as well, and in two different ways. \(328=83^2 - 81^2 = 43^2 - 39^2\)<br /><hr style="font-family: inherit;" /><b>The 329th Day of the Year</b><br />329 = 7 x 47, the sixth of the last nine numbers that is a semi-prime.</span></span></div><div><span><span style="background-color: white; color: #222222;"><br />329 is not divisible by 3, 2, or 9, but on the bright side, the sum of all its proper divisors is a palindrome, 55.</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;">329 is the sum of three consecutive primes. 107 + 109 + 113</span></span></span></div><div><span><span style="background-color: white; color: #222222;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;"><br /></span></span></span></div><div><span><span style="background-color: white; color: #222222;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;">329 is the number of forests (trees and disconnected trees) possible with ten vertices.</span><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;">329 is also a happy number, since the iteration of the sums of the squares of its digits includes 1, </span></span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">329 is another value of n for which n^2 + n + 1 is prime. Computer people, how many year days are like 1, 2, 3, and 329 and make that quadratic prime?</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">Teachers, a great day to remind your students why a 4n+1 day like 329 is not expressible as the sum of two squares.</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">329 is expressible as the sum of three squares in three unique ways, \(18^2 + 2^2 + 1^2 = 17^2 + 6^2 + 2^2 = 10^2 + 15^2 + 2^2 =16^2 + 8^2 + 3^2 = 329 \) , and more ...</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">Like all odd number, 329 is the difference of consecutive squares, \(165^2 - 164^2\), but it is also \(27^2 - 20^2\). <br /><hr style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><b>The 330th Day of the Year</b></span></span></div><div><span><span style="background-color: white; color: #222222;">330 = 2 x 3 x 5 x 11, and the sum of six consecutive primes, 43 + 47 + 53 + 57 + 61 + 67, and of five consecutive squares </span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;"><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;">If all the diagonals of an eleven sided regular polygon were drawn, they would have 330 internal intersections.</span><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;">330 is the last year day which is a pentagonal number. It is the sum of fifteen consecutive integers starting with the integer 15. (All Pentagonal numbers follow a similar pattern) The average of all the pentagonal numbers up to 330 is the 15th triangular number.</span><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><br style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="font-size: 13.2px;">A set of 11 points around a circle provide the vertices for 330 quadrilaterals. Thus 11 choose 4 = 330</span></span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">330 is divisible by the sum of its digits, making it a Harshad, or Joy-Giver number.</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">The sum o 330 raised to each of its digits in turn, is a prime, \(330^3 + 330^3 + 330^0\) is prime. *Prime Curios</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">330 = 303 + 3^3 *Derek Orr</span></span></div><div><span><span style="background-color: white; color: #222222;"><br /></span></span></div><div><span><span style="background-color: white; color: #222222;">And who would have thought, there are 330 dimples on a British Golf Ball, *Derek Orr, </span></span></div><div><span><span style="background-color: white; color: #222222;">It seems most US golf balls have 336, but some manufacturers have as many as 360, and one has as many as 1120. <span face="Roboto, sans-serif" style="color: #444444; font-size: 16px;">the 2017/18 model of the popular </span><span face="Roboto, sans-serif" style="color: #444444; font-size: 16px; font-weight: 700;">Titleist Pro V1</span><span face="Roboto, sans-serif" style="color: #444444; font-size: 16px;"> has 352 dimples on it, (Think they will send me a case?)</span></span></span></div><div><span><span style="background-color: white;"><span face="Roboto, sans-serif" style="color: #444444;"><br /></span></span></span></div><div><span><span style="background-color: white;"><span face="Roboto, sans-serif" style="color: #444444;">330 is the sum of three squares in multiple ways, \(4^2 + 5^2 + 17^2 = 5^2 = 7^2 + 16^2 = 330\)</span><br /></span></span></div><div><span><span style="background-color: white; color: #222222;"><hr style="font-family: inherit;" /><br /></span></span></div></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-37539628762276597492020-08-18T17:35:00.010-07:002021-05-30T14:17:49.012-07:00Number Facts for every Year Date, 271- 300<b>The 271st Day of the Year</b> <br />271 is the 58th prime number, the larger of a pair of twin primes, and is the sum of eleven consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43).<br /><br />271 is also the difference of two consecutive cubes, 10<sup>3</sup> - 9<sup>3</sup>. Such prime numbers are called Cuban Primes, and were named by the British mathematician Allan Joseph Champneys Cunningham in 1923. <br /><br />Using the English alphabet code, a = 1, b = 2, etc, there are exactly 271 positive numbers that give larger numbers when you write out their English names and add the letters *primecurios<br /><br />271 is the first three digits of e=2.718281828459045....<br /><br />Prime Curios states that the largest prime factor of any five digit repdigit, is 271. <br /><br />271^2 = 73441, and its reversal, 14437 is prime, and 271^3 = 19902511, and its reversal, 1150991 is also prime. *Prime Curios And now you wonder???<br /><br />271 is the smallest number between numbers with cubes for factors. 270 = 2 x 3^3 x 5, and 272 = 2^4 x 17<br /><br />Of all the primes between 2 and 269, none of them has period or five for their reciprocal. 1/271 = .00369..., which explains why all five digit repdigits are divisible by 271. <br /><br />271 is a 4n+3 prime, so it is not the sum of two squares. It is however, the difference of two consecutive squares, 136^2 - 135^2. Of the six permutations of the digits of 271, four of them are centered hexagonal number. Since they must always be odd, you can quickly figure out which two are omitted. <br /><br />271 = 15+16^2 (n + (n+1)^2). Counting 0+1^2, it is the 16th such number. <br /><hr /><b>The 272nd Day of the Year</b> <br />272 = 2<sup>4</sup>·17, and is the sum of four consecutive primes (61 + 67 + 71 + 73). <br /><br />272 is also a Pronic or Heteromecic number, the product of two consecutive factors, 16x17 (which makes it twice a triangular #). <br /><br />And 272 is a palindrome, and the sum of its digits, 11, is also a palindrome. (can you find the next?)<br /><br />272 = 16^2 + 4^2 and the difference of two squares 69^2 - 67^2 and also 36^2 - 32^2 , and 21^2 - 13^2<div>And speaking of that last one, a fun Fibonacci fact, the difference of the squares of consecutive Fibonacci numbers, is the product of the one before times the one after, 8 x 34 = 272. This is true for all differences of consecutive Fibonacci numbers greater than one. <br /><br />272 = 24·17, sum of four consecutive primes (61 + 67 + 71 + 73)<br /><br /> There are 272 ways to partition 39 into prime parts. <br /><br />272 can be written as the sum of distinct powers of 4, 272 = 4^4 + 4^2, <br /><br />All the digits of 272 are primes, and the sum of the digits is prime <br />272 is divisible by four, and so 272 = 69^2 - 67^2. It is also divisible by eight, and so it is = 36^2 - 32^2, and it is also divisible by 16 so, 21^2 - 13^2 <hr /><b>The 273rd Day of the Year</b> <br />273<sup>o</sup>K(to the nearest integer)is the freezing point of water, or 0<sup>o</sup>C <br /><br />OOOOH wait, 273 = 13*7*3, and 1373 is also prime.. and There are only two sphenic numbers consisting of concatenation of distinct prime numbers, this is the smaller of the two.(sphenic or wedge numbers are products of three distinct primes) *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime curios</a><br /><br />273 is a repdigit in hexdecimal, or base 16 (111) 16^2 + 16 + 1, and in base 20(vigesimal), where it looks like a bad report card (DD)<br /><br />273 = 47^2 - 44^2 and also 137^2 - 136^2<br /><br />Prime Curios says that 273^10 - 10^273 is the smallest n for which this expression is prime. There are no more year days that exhibit this relationship. <br /><br />Prime Curios includes this tasty factoid, the prime factors of 273 = 3 x 7 x 13, and if concatenated in reverse order, 1373, it is prime. <br /><br />273 is a Palindrome in base 2(100010001). 2^8 + 2^4 + 2^0. Also a Palindrome in base 4 (10101) , 4^4 + 4^2 + 4^0. <br /><br />273 is the product of two consecutive Fibonacci numbers (13 x 21) and is thus a Golden rectangle Number. Each of these as they gown larger approximate a Golden Rectangle. 21/13=1.615.... <br /><br /><hr /><b>The 274th Day of the Year</b><br />274 is a tribonacci number.The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are 0, 0, 1, 1, 2, 4, 7,..Are there more Tribonacci number in the year Days? <br /><br />274 is also the sum of five cubes, 2<sup>3</sup> + 2 <sup>3</sup> + 2<sup>3</sup> + 5<sup>3</sup> + 5<sup>3</sup>, and of three triangular numbers 78 + 91 + 105. They are consecutive triangular numbers because their differences are consecutive numbers, 91-78 = 13 and 105 - 91 = 14. In 1796, Gauss proved that every positive integer could be expressed as the sum of three triangular numbers. <br /><br />274 is an example of Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity) 274= 2 * 137 and 2+ 7 +4 = 13 = 2 + 1 + 3 + 7. Find another.<br /><br />Sort of strange one from Prime Curios, "(274 - 2)2 + (274 - 7)7 + (274 - 4)4 is prime." Who would even think to check such things? (Impressive curiosity!) <br /><br />The centered Triangular numbers start with a single point, and then build a triangle around it with side lengths of 1, then 2, etc. I mention this because 274 is a centered triangular number, The difference between two centered triangular numbers increases by three each time. CT(n)= 3n(n-1)/2 + 1. 274 is the 14th centered triangular number. <br /><br />137 = 11^2 + 4^2. Can you see how that leads you to 274 = 15^2 + 7^2 because it is twice a number which is the sum of two squares. <br /><hr /><b>The 275th Day of the Year</b><br />275 = 138^2 - 137^2 = 30^2 - 25^2<br /><br />275 is the number of partitions of 28 in which no part occurs only once. (Students might try finding the similar number of partitions for 10, or some smaller number to get a sense for how they grow)<br /><br />275 is the maximum number. of pieces that can be formed from an annulus with 22 straight lines. <br /><br />Imagine taking the number 11 and forming every possible partition of it, and then you count all the parts of all these partitions, you get 275, but a shortcut is to take the largest number in each partition, and add those.... Yep, same number.(shortcuts anyone?). Students might try that with 2,3,4 etc and see if they can figure out why they are the same.<br /><br />275 is only the fifth number that is expressible as 2^n + 3^n. 275 = 2^5 + 3^5. <br /><hr /><b>The 276th Day of the Year</b> <br />276 is the sum of twelve consecutive prime numbers (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43).<br /><br />It's also the sum of three consecutive fifth powers, 1^5 + 2^5 + 3^5 = 276<br /><br /> And from the trivia file, 276 is the number of rounds of the longest boxing match in history. Jack Jones beat Pat Tunney in a bare-knuckle fight in 1825 after 4 1/2 hours.<br /><br />If we let S(n) = the sum of the proper divisors of n (so S(8) = 1 + 2 + 4 = 7) and make a chain S(n1) = n2 and s(n2) = n3... eventually you get back to a number you already had. Some numbers (called perfect numbers) do so immediately. Others form chains of 2 (sociable numbers) or 4 (the most common) or more, but every number will loop back........ almost. It seems 276 is a maverick and just keeps getting bigger all the time.... No other number is known that does not form a chain. <br /><br />A Four by Four Magic square using only consecutive prime numbers by A. W. Johnson, Jr.. What's the magic constant??? What day is it?<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-FrFSDEHqKP0/XytPDml3xNI/AAAAAAAANAo/xA0U0Ge0tew0S9SXnDiYVW_OJRgc4RqUQCLcBGAsYHQ/s1600/magic%2Bsquare%2Bconsecutive%2Bpimres%2B276.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="154" data-original-width="240" height="257" src="https://2.bp.blogspot.com/-FrFSDEHqKP0/XytPDml3xNI/AAAAAAAANAo/xA0U0Ge0tew0S9SXnDiYVW_OJRgc4RqUQCLcBGAsYHQ/s400/magic%2Bsquare%2Bconsecutive%2Bpimres%2B276.jpg" width="400" /></a></div><br />See 258 for a smaller also using consecutive primes. <br /><br />276 = 70^2 - 68^2<br /><br />276 is a palindrome in base 8 (424) and a repdigit in base 22(CC)<br /><br />And 276 is a triangular number, the sum of the first 23 integers. <br /><hr /><b>The 277th Day of the Year </b><br />277 is the 59th prime number, It is also a self number, and is the largest prime self number that can be a day of the year (A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 20, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar.... The next prime self number is 367, too large to be the number of a day of the year)<br /><br />The ever-clever Derek Orr pointed out, "Keep going, 277--59th prime, 59--17th prime, 17--7th prime." And "277 = 1<sup>5 </sup> + 1<sup>5 </sup> + 2<sup>5 </sup> + 3<sup>5 </sup> (first four Fibonacci numbers raised to the next Fibonacci number(5) power. What would be next?)"<br /><br />277 is also a Pythagorean Prime. And as Fermat wrote, all 4n+1 primes are the sum of two squares in only one way. Every prime of the form p = 4n+1 can be the hypotenuse of an integer sided right triangle. In this case the triple is (115, 252, 277) Note all the permutations of two alike one different. <br /><br />The reciprocals of the 59 primes from 2 to 277, 1/2 + 1/3 + 1/5 + 1/7 + .... + 1/277, is barely larger than 2, the smallest primes. *Prime Curios <br /><br />The square of 277 is also the sum of two squares (115, 252, 277) (Note that all three numbers are two of one digit of another, and they have all three orders of two alike, one different. <br /><br />277 = 277 = 139^2 - 138^2<br /><br />277 is the largest factor of the #7(the product of the first seven primes) + 1. Some write this notation as \( p_7#\), perhaps even clearer, would be P!(7) for prime factorial. <br /><br />a flat disk can be cut into 277 sections with only 23 straight cuts. <br /><br />On an infinite chessboard, there are 277 squares a knight can reach in six moves. (I wonder how many he has touched along these 277 trips?)<br /><br />277 in Roman numerals is sort of Palindromic in that the characters CLVI appear in groups of 21212, CCLXXVII<br /><br />277 is also a palindrome in base 12, the Duodecimal system. (1B1)<br /><br />277 = 4^4 + 4^2 + 4^1 + 4^0. <br /><hr /><b>The 278th Day of the Year</b><br />278 = 2x2x2x2x2x2x2x2+22<br /><br />1789 - 278= 1511 is prime (278th prime minus 278).<a href="https://www.blogger.com/*http://derektionary.webs.com" target="_blank">Derek's Daily Math </a>for more<br /><br />278 is the fifth smallest number n such that n<sup>n</sup> starts with the digits of n, \(278^278 =2.78... × 10^679 \) <br /><br />If you like the game "Brussels Sprouts", then you should know that the total number of moves is always a number that is three more than a multiple of five, like 278> 277 = 14^2 + 9^2 = 139^2 - 138^2<br /><br />There are 278 primes less than 1800. <br /><hr /><b>The 279th Day of the Year</b><br />Every positive integer is the sum of at most 279 eighth powers. See <a href="http://en.wikipedia.org/wiki/Waring%27s_problem" target="_blank">Waring's problem</a> <br />279 seems to require 24 itself, 2^8 with twenty-three 1^8. <br /><br />279 = 3^2 + 3^3 + 3^5 (powers are consecutive primes). *Derek Orr It is also 7^3 - 4^3 (PB)<br />For more fun with daily calendar math see <a href="http://derektionary.webs.com/days-of-the-year" target="_blank">Dereks Daily Math</a> <br />and <a href="http://benvitalenum3ers.wordpress.com/" target="_blank">Ben Vitale</a><br /><br />Another from the prolific Derek Orr, 279 + 10^57 (this means there is a prime gap of more than 279 between this prime and the one before it. 10^57 + 279 is the first prime with 58 digits.) <br /><br />279 = 1^2 + 2^2 + 7^2 + 15^2, It can not be done with fewer positive squares. (But is that the ONLY sum of four squares?) <br /><br />279 = 48^2 - 45^2, and also 140^2 - 139^2<br /><br />279 = \(8!_2 - 7!_2\) This is me persisting against the use of n!! for the alternate factorial, 8 x 6 x 4 x 2.<br /><br />And Derek Orr has 279 is a divisor of 999,999,999,999,999. If I understand him, it will not divide any shorter string. <br /><hr /><b>The 280th Day of the Year</b> <br />The 280th day of the year; There are 280 plane trees with ten nodes. As a consequence of this, 18 people around a round table can shake hands without crossing arms in 280 different ways (up to rotations) <br /><br />The sum of the first 280 consecutive primes mod 280 is prime. *Prime Curios (Stijn Dierckx @Stanny1990 told me there are 108 such days in a year [almost 30% of the days satisfy this property]. Next one in 5 days!)<br /><br />280 = 71^2 - 69^2 = 37^2 - 33^2 = 4^3 + 6^3 <br /><br />18 people around a round table can shake hands with each other in non-crossing ways, in 280 different ways (this includes rotations) *Wikipedia <br /><br />280 is a palindrome in base 3(101101)<br /><br />280+ 1 is prime, and 280^2 + 1 is prime *Derek Orr<br /><br />\(280 = 10!_3\) = 10 x 7 x 4 x 1 <br /><br />And 280 is a Joy-Giver (Harshad) number, divisible by the sum of its digits. <br /><hr /><b>The 281st Day of the Year </b><br />281 is the 60th Prime, and a smaller twin prime and is the sum of the first fourteen prime numbers. It is also the sum of 7 consecutive primes starting at 29<br /><br />Prime Curios offers that 281 is the larges prime such that the (sum of the factorials from 1 to P ) - 2 is prime <br /><br />281 is the sixth, and last, day of the year such that the sum of the first k primes is a prime. (<i>I just noticed that all of them except 2, is the smaller of pair of twin primes. Unfortunately, the next one after that is not.</i>)<br /><br />281 appears in the sequence of primes generated by \(f(x)= x^2 + x + 41 \) Which is often called the Euler Polynomial. (although Euler actually used \(x^2 -x + 41\) which is prime for x values from 1 to 40. Legendre noticed that the positive x form produced the same primes for values from 0 to 39.)<br /><br />Here are four squares with the same digits 281<sup>2</sup> =78961, 286<sup>2</sup>=81796, 137<sup>2</sup>=18769,133<sup>2</sup> = 17 689.<br />Since five unique digits can represent 120 different numbers, how many, on average, are squares?<br /><br />Found this one in the Twitter feed of Jim Wilder@wilderlab in 2016, For day 281, a palindrome: 281=9•8+7•6+5•4+3•2+1+2•3+4•5+6•7+8•9<br /><br />281 is a prime of the form\(10!_3 + 1\).<br /><br />If you find 281!, then remove all the Primes (divide by p#281), that value +/- 1 are twin primes.<br /><br />Another from Prime Curios, There are 281 triangular numbers on the digital clock, including seconds. What's the first one after midnight. <br /><br />281 is the smallest prime such that the period length of the reciprocal is (p-1)/10. <br /><br />281 is a Pythagorean Prime, the hypotenuse of (160, 231, 281). <br /><br />And since 281 is a 4n+1 prime, it is the sum of two squares, 281 = 16^2 + 5^2, and because it is an odd number, 281 = 141^2 - 140^2<br /><hr /><b> The 282nd Day of the Year</b> <br />there are 282 plane partitions of nine objects. (A plane partition is a two-dimensional array of integers n_(i,j) that are nonincreasing both from left to right and top to bottom and that add up to a given number n.) (That reads much harder than the idea, here is an image of a plane partition of 22 from Mathworld, which, as they say, is worth a thousand words: <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-D1jE1ipphGg/V_GRhyuFtMI/AAAAAAAAIfg/hAdOn78txHsMOd7etsRshPdnwmfMR3UmwCLcB/s1600/plane%2Bpartition%2Bof%2B22.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-D1jE1ipphGg/V_GRhyuFtMI/AAAAAAAAIfg/hAdOn78txHsMOd7etsRshPdnwmfMR3UmwCLcB/s1600/plane%2Bpartition%2Bof%2B22.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br /><br />282 is the smallest number between twin primes which is a palindrome. Can you find the next?<br /><br />282 is the largest gap between two successive primes below one billion.<br /><br />There is no digit you can place after 289 which makes the result a prime. *Derek Orr <br /><br />The sum of the divisors of 282 is 576, a square number. It is the 15th such number this year, and there will be six more.<br /><br />And sitting between twin primes, 282 is the average of two consecutive primes. <br /><br />282 is a palindrome in base 10 (282) and a repdigit in base 9 (333)<br /><br />282 can be written as the sum of nine positive fifth powers, but it takes twelve positive fourth powers. <br /><br />An admirable number is a number so that s(n) (the sum of the divisors of n) - 2d (for some divisor d of n)= 2n. For 282 that works with d=6. 576 - 12 = 564 =2 x 282. <br /><br />The 282nd triangular number is the first triangular number divisible by 47 <br /><hr /><b>The 283rd Day of the Year </b><br />5 + 8<sup>1</sup> + 3<sup>5</sup>. and one more way below<br />It is the 61st Prime, which makes it a prime of prime order. The sum of its digits is also a prime. And it is the concatenation of two primes 2 and 83.*Derek Orr<br /><br />And 6 x 283 sits between another pair of twin primes. (PB) <br /><br />283 can be expressed as n<sup>n</sup> + (n+1)<sup>n+1</sup> (Find n.) The largest prime year day for which this is true.<br /><br />283 = 3^3 + 4^4. *Derek Orr <br /><br />Like all odd numbers, it is the difference of consecutive squares, 283 = 142^2 - 141^2.<br /><br />283 = (6! - 5! - 4! - 3! - 2! - 1! - 0!)/2.<br /><br />283 in base eight is 238, a permutation of its own digits.<br /><br />283 is a prime of the form 4n+3, Bernard Frénicle de Bessy discovered that such primes cannot be the hypotenuse of a Pythagorean triangle (1676), as opposed to primes of the form 4n+1, which Fermat conjectured always were in 1640.<br /><br />The smallest amicable pair is (220, 284). 283 is the only prime adjacent to either of them, making it the smallest prime adjacent to an amicable number, and the only such prime that is a year day. There are no primes adjacnt to the next smallest pair.<br /><br />Prime Cuios says the state of Pennsylvania is 283 miles, west to east. <br /><hr /><b>The 284th Day of the Year</b><br />The 284th day of the year; 284 is an amicable (or friendly) number paired with 220. The divisors of 220 add up to 284 and the divisors of 284 add up to 220. Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by Thābit ibn Qurra (826-901).(Can you find the next pair?)<br /><br />jim wilder @wilderlab pointed out a variation of friendly numbers in degree three... "Along the lines of friendly numbers... 136 = 2³ + 4³ + 4³ and 244 = 1³ + 3³ + 6³"<br /><br />284 is divisible by four, so it is the difference of two squares, 72^2 - 70^2. <br /><br />Lagrange's theorem tells us that each positive integer can be written as a sum of four squares. This theorem allows for some squares of zero. If you restrict the set to only those that require four positive squares for the sum is much smaller. 284 is such a number, and is the 46th such number. 289 = 17^2 + 4^2 + 2^2 + 2^2. It seems that there can sometimes be consecutive integers that require four squares (239,240), but never a triplet.<br /><br />The 51st prime is 231, making 284 the sum of 51 and the 51st prime. <br /><br />And 284 is a palindrome in base 8(434) <br /><hr /><b>The 285th Day of the Year</b> <br />285 is a square pyramidal number (like a stack of cannonballs, or oranges with the base in a square)... Or.. the sum of the first nine squares. \( 285 = \sum _{i=1} ^ 9 (i^2) \). 285 is the largest Year day that is a Square Puramidal number. <br /><br />285 is 555 in base 7. <br /><br />285 = 3^2 + 5^2 + 7^2 + 9^2 + 11^2<br /><br />Not sure how rare this is, but just saw it on MAA's Number a Day and was intrigued, 285^2 = 81225 uses the same digits as 135^2 (18225) and 159^2 (25281).<br /><br />285 is a Joy-giver (Harshad) number, divisible by the sum of its digits. <br /><br />285 is the difference of two squares, 285 = 31^2-26^2 = 49^2 - 46^2. <br /><br />A unusual relation to produce the Square Pyramidal numbers, For odd n, (n+n+2+c)^3 - n^3 - (n+2)^3 - (n+4)^3 divided by 524 it gives the nth square Pyramidal number. When n=7, you get 285.<br /><br />285 is the longer leg of a primative Pythagorean Triple. (68, 285, 293) <br /><hr /><b>The 286th Day of the Year</b> <br />286 is a sphenic (wedge) number, the product of three distinct primes, 2 x 11 x 13. <br /><br />If you take the product of each pair of numbers that add up to 12, their sum will be 286. There is a similar pattern for every triangular pyramidal number. <br /><br />286 is a tetrahedral number (<i>a triangular pyramid, note that 285 was a square pyramidal number, how often can they occur in sequence?</i>) It is the sum of the first eleven triangular numbers, 286 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 <br /><br />And to top yesterday's curiosity, here are four squares with the same digits 286<sup>2</sup>=81796, 137<sup>2</sup>=18769, 133<sup>2</sup> = 17 689, 281<sup>2</sup> =78961 <br /><br />286 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 + 11^2 = *Derek Off, It is the last year day which is sum of squaes of first n primes.<br /><br />286= 7^3 - 7^2 - 7^1 -7^0 *Derek Orr<br /><br />There are factorials, lesser known subfactorials, and an even more obscure Swinging Factorial. The swinging factorial. The symbol is a backwards s curve. But I'll use n!S. it is given by n! divided by the square of the factorial of the floor function of n/2. The function osscillates up and down after the first few terms. 5!S for instance is is 5!/(2!)^2 = 120 /4 = 30. 6!S is 6!/(3!)^2 = 720/36 = 20 .Explore........ <br /><hr /><b>The 287th Day of the Year </b><br />287 = 7^3 - 7^2 - 7^1. *Derek Orr<br /><br />287 is not prime, but it is the sum of three consecutive primes (89 + 97 + 101), and also the sum of five consecutive primes (47 + 53 + 59 + 61 + 67), and wait, it is also the sum of nine consecutive primes (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). *Prime Curios<br /><br />287 is the smallest non-prime Kynea number and the 4th overall.A Kyneaa number is an integer of the form 4<sup>n</sup> + 2<sup>n+1</sup> − 1, studied by Cletus Emmanuel who named them after a baby girl. The binary expression of these numbers is interesting, 287<sub>[2]</sub> is 100011111. Each Kynea number has a one, followed by n-1 zeros, followed by n+1 ones. The Keyna primes are all two less than a square number. There are four year days that are Kynea numbers (7, 23, 79, 287), but 287 is the only one that is composite.<br /><br /><div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-iNJVQOBGypI/XzQHDK4qq4I/AAAAAAAANBc/hr-rXkJF_38iT6yl5XNOTf3su7l33ORDQCLcBGAsYHQ/s507/pentagonal%2Bnumbers%2Bimage.png" style="display: block; padding: 1em 0px;"><img border="0" data-original-height="95" data-original-width="507" src="https://1.bp.blogspot.com/-iNJVQOBGypI/XzQHDK4qq4I/AAAAAAAANBc/hr-rXkJF_38iT6yl5XNOTf3su7l33ORDQCLcBGAsYHQ/s0/pentagonal%2Bnumbers%2Bimage.png" /></a></div>287 is the 14th Pentagonal number. Pentagonal numbers are figurate numbers, but unlike the triangular and square numbers, they are not rotationally symmetric. Instead each starts at the same first point and overlaps previous ones. The number in each outer level is a multiple of five, but the total in each figure is given by \( P(n) =\frac {3n^2 - n}{2} \). There is a generalized Pentagonal number which is important in Euler's theory of partitions. It extends the domain to zero and negative numbers. <br />There are only six numbers that can not be expressed as the sum of four or less pentagonal numbers. They are all less than 100. Can you find them?<br /><br />Derek Orr points out that 287^2 + 287 + 1 = 82657 is prime.<br /><br />287 is another number that require four positive squares for their sum. 287 = 15^2 +7^2 + 3^2 + 2^2.<br /><br />287 is only the fifth number that is an RMS (Root-Mean-Square) Number, the RMS of the divisors is an integer. RMS(287) = sqrt((1^2 + 7^2 + 41^2 + 287^2)/4)=145<br /><br />Partitions of 40 include ways to select one or more of those numbers that sum to 40. 5+5+10+20 =40, and for that example, it turns out (not by chance) that the sum of their reciprocals, 1/5 + 1/5 + 1/10 + 1/20 = 1, and integer. Now you find one, because according ot OEIS, there are 287 such partitions. Happy Hunting! <br /><hr /><b>The 288th day of the year</b>; <br />288 is the super-factorial of four. 1! x 2! x 3! x 4! =288. It is important that math students learn not to say this number in public as it is two gross.<br /><br />288 is also the sum of the first four integers raised to their own power. 288 = 1^1 + 2^2 + 3^3 + 4^4<br /><br />288 = 2^3 + 4^3 + 6^3 <br /><br />288 is the smallest non-palindrome, non-square, that when multiplied by its reverse is a square: 288 x 882 = 254,016 = 504^2 <br /><br />288^2 + 288 =/- 1 form a pair of twin primes. <br /><br />288 is the 8th Pentagonal Pyramid number, 1 + 5 + 12 + 22 + 35 + 51 + 70 + 196. , <br /><br />288 is a Joy-Giver (Harshad) number, divisible by 18, the sum of its digits. <br /><br />288 = 2^5 x 3^2, Such numbers with every prime factor to the second, or higher integer is called an Achilles Number.<br /><br />288 in base 3, 4, 6, and 12 all end in 200. <br /><br />288 has many factors of two, producing several different expressions as difference of squares, 288 = 73^2 - 71^2 = 38^2 - 34^2 = 22^2 - 14^2 = 17^2 - 1^2 <br /><br />22 can be written as the sum of four squares (including zeros) in 288 ways. 288 = 2^3 + 4^3 + 6^3 <br /><br /><hr /><b>The 289th Day of the Year </b><br />289 = 15^2 + 8^2<br /><br />289 is the Hypotenuse of a Primetime Pythagorean Triangle (161, 240 289), which means 289^2 is also the sum of two squares.<br /><br /></div><div>There are only two square year days that are neither the sum, or the difference of two primes, 289 is the second. </div><div><br /></div><div>The 289th day of the year; 289 is a Friedman number since (8 + 9)2 = 289 (A Friedman number is an integer which, in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷) and sometimes exponentiation.)Students might try to find the first few multi-digit Friedman numbers.<br /><br />289 is the square of the prime 17, and the square of the sum of the first four primes, 289 = (2 + 3 + 5 + 7)^2 <br /><br />289 is the largest 3-digit square with increasing digits(amazing, to me, that this occurs so early in the 3-digit numbers.)<br /><br />289 is the hypotenuse of a primitive Pythagorean triple. Find the legs students!<br /><br />289 is a 4n+1 prime, which Fermat told us must be the sum of two squares in exactly one way, 8^2 + 15^2 = 289. <br /><br />As an odd number, it must also be the difference of two consecutive squares, 145^2 - 144^2 = 289. <br /><br />Derek Orr points out this pattern about 289: <br />289 = 17^2.<br />289 in base 4 = 10201 = 101^2.<br />289 in base 8 = 441 = 21^2.<br />289 in base 14 = 169 = 13^2.<br />289 in base 15 = 144 = 12^2.<br />289 in base 16 = 121 = 11^2.<br />289 in base 17 = 100 = 10^2.<br />289 in base 36 = 81 = 9^2.<br /><br />289 is one more than twice a square, 2 x 12^2 = 17^2 = 289. Such numbers lead to good approximations of \(\sqrt{2}\). 17/12 = 1.41666... The equation 2x^2 + 1 = y^2 is called a Pell Equation (although only because Euler mis-attributed it.)<br /><br />The sum of the divisors of 289, 1+17+289 = 307, a prime.It is only the 7th such number, and the last that is a year day.<br /><br />There are only two year days which are squares and which are neither the sum, nor the difference, of two primes. 289 is the larger, The other is also a three digit number, and a palindrome.</div><div><br />289 is the sum of 4 non-zero 4th powers. 1^4 + 2^4 + 2^4 + 4^4. <br /><hr /><b>The 290th Day of the Year </b><br />290 = 13^2 + 11^2 = 17^2 + 1^2.<br /><br />290 is a sphenic (wedge) number, the product of three distinct primes (290 = 2*5*29).<br /><br />It is also the sum of four consecutive primes (67 + 71 + 73 + 79) [Students might try to construct and examine a list of numbers that can be written as the sum of two or more consecutive primes]<br /><br />Derek Orr Pointed out that there are 290 primes below 1900<br /><br />290 is conjectured to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome.<br /><br />290 is the tenth prime(29) times ten<br /><br />290 in base 3 is the concatenation of two three digit palindromes, [101202], and a singl 3-digit palindrome in base twelve, and in base 3, 4, 6, and 12, they all end in the digits202. <br /><br />290^3 - 290^2 = 4930^2. 290 is only the 17th number for which n^3 - n^2 = y^2. There will only be two more year days that meet that relation and each is one more than a square. And, the sum of the squares of the divisors of 17 is 290.<br /><br />290^2 + 290 +/- 1 form twin primes. *Derek Orr<br /><br />290 = 1^4 + 1^4 + 2^4 + 2^4 + 4^4. Can it be done with fewer fourth powers? <br /><hr /><b>The 291st day of the Year</b><br />291 is the largest number that is not the sum of distinct non-trivial powers.<br /><br />ϕ(291)=192 The number of integers less than, and relatively prime to 291 is equal to it's reversal, 192.<br /><br />291 is also equal to the nth prime + n.... but for which n, children?<br /><br />The sum of the aliquot divisors of 291 is prime, 101. *Derek Orr <br /><br />291 appears in five pythagorean triangles, one as the hypotenuse (195, 216, 291) and four more as the shorter leg <br /><br />291 is a palindrome in base 9(353), and a sequence of three consecutive integers in base 16(123)<br /><br />291 is a Happy Number, 2^2 + 9^2 + 1^2 = 86, 8^2 + 6^2 = 100, 1^2 + 0^2 + 0^2 = 1. Happy, Happy, Happy!<br /><br />Because 291 = 6 x 47 + 9, it can be written as the difference of two squares, 291 = 50^2 - 47^2. (An interesting note is that every power of 2^n is either one more or one less than some member of the 6n+9 sequence. The converse is not true, 21 is easy counterexample. <br /><br />And the 291st digit of pi is a zero. <br /><hr /><b>The 292nd Day of the Year</b> <br />292 = 74^2 - 72^2, and also 16^2 + 6^2<br /><br />The continued fraction representation of pi is [3; 7, 15, 1, 292, 1, 1, 1, 2...]; the convergent obtained by truncating before the surprisingly large term 292 yields the excellent rational approximation 355/113=3.4151929 for pi. (113355, divide in the middle, put big over little)<br />The approximation was found by Chinese mathematician and astronomer Zu Chongzhi(429–500 AD), using Liu Hui's algorithm which is based on the areas of regular polygons approximating a circle*Wik<br /><br />292 is the number of ways to make change for 1 dollar (or for 1 Euro), using only 1, 5 and 25 cent coins (base five coins).<br /><br />292! + 291! ± 1 are 595-digit twin primes. (Can you find smaller sums of consecutive factorials like this that are twin primes?) <br /><br />292 is a palindrome in base 10, base 8(444), and base 7(565), and in base 2(100100100) it repeats the three digits 100, three times (which should explain the 444 in base eight. <br /><br /><hr /><b>The 293rd Day of the Year </b><br />293 = 17^2 + 2^2 = 147^2 - 146^2, <br /><br />293 is the Hypotenuse of a Primitive Triple, (68, 285, 293). <br /><br />293 is a Sophie Germain Prime. (A prime number p such that 2p + 1 is also prime.) Sophie Germain used them in her investigations of Fermat's Last Theorem. It is an unproven conjecture that there are infinitely many Sophie Germain primes.<br /><br />293 is also the sum of five cubes, 293=2^3 + + 2^3 + 3^3 + 5^3 + 5^3<br /><br />and from Jim Wilder @Wilderlab : 293^202 begins with the digits 202 and 202^293 begins with the digits 293.<br /><br />Prime Curios points out that 300 is kind of prissy, but for math bowlers, the largest prime you can get in the game is 293. Just got to pick off three in the corner on this last roll....<br /><br />In a normal (non leap year) there are 293 non-prime days in a year. <br /><br />Another frome Derek Orr, 239 + 2*3*9 = 293. <br /><br />More from Orr, he is on a roll today: 17 is a prime, and 17^2 + 4 = 293. 293 is a prime, and 293^2 + 4 = 85853, and it is prime. Stop! Go no farther. Only disappointment awaits. <br /><br />293 is another Happy number, the iteration of the sum or the squares of the digits, arrives at one. <br /><hr /><b>The 294th Day of the Year</b> <br />292^2 + 293^2 + 294^2 is prime. *Derek Orr <br /><br />294 is a practical number because all numbers strictly less than 294 can be formed with sums of distinct divisors of 294. There are only 84 such numbers in the year.<br /><br />294 is the sum of four consecutive squares, 7^2 + 8^2 +9^2 + 10^2 <br /><br /> Just learned about "digitally delicate primes," although they are still pretty new in general. In 1978, the mathematician Murray Klamkin wondered if there were any primes so that if you change any digit to any other digit, the newly formed number would be composite. </div><div> 294 is NOT such a number, heck it is not even prime, and there is no year day that is a digitally delicate prime. The smallest one is 294001. If you change any one of these six digits to anything other than what it is you can produce 9^6 = 531,441 different numbers share five of these digits in their present location, and ALL OF THEM ARE COMPOSITE. Change the 2 in front to a 1, 194001, and of course that is divisible by 3, so try a 3 in front, 394001. That is 47×83×101. Change the first 0 to 2 to get 294101 = 19×23×673 . <br /> The fact is, there seems to be an infinite number of these "digitally delicate darlings." <br />If that isn't extreme enough for you, somebody decided to extend that to something called "widely digitally delicate" primes. It seems they have somehow proved those are infinitely available too, but I can't tell you the smallest one because, well, nobody has ever found one. (Ok, I'm not sure how they prove there is an infinite number of something that they can't find even one of, but the big brains in math seemed to agree they have. The way they work is that if you imagine an infinite number of zeros in front of a prime, and you change any one of those zeros to something else, "ca-ching", it's composite. <br /> 294001 is not one of them, I checked. If you change the zero in front of 0294001 to anything else, it's composite. But if you change the NEXT zero to get 10294001.... bingo, a prime, so the digitally delicate 294001 is NOT WIDELY digitally delicate.</div><div> <br />Some primes are called unique because no other prime has the same period for its reciprocal (There are only 23 of these known for primes below 10^100. None of these primes contain more than eight different digits. Of all those primes with 8 different digits, has period 294. Notice the similarity to the period of 7 (142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143)*Prime Curios, *Wikipedia<br /><br />Derek Orr shared that there is no prime in the decade of numbers between 294x10 and 295x10<br /><br />Found this oddity in my notes: 111152- 2942 = 123,456,789<br /><br />Quick arithmetic note. Any number evenly divisible by six is the sum of three consecutive integers, 294 = 97+98+99<br /><br />And I'm writing this in a leap year, 294 + (2x9x4) =366<br /><br />2(294)+9(294)+4(294) - 1 is 4409, a prime<br /><br />294 is the only year day (and the only number I know of) which can be written two different ways as the sum of prime numbers using only the digits 1-5 exactly once in the digits of the prime: 294 = 43 + 251 = 53 + 241. (And the largest year day that can be written in this manner in a single way, is a permutation of the digits of 294, 249.) Students might try to find all the numbers that can be written as the sum of primes using digits 1-6 only once each.<div><br />Consider the reciprocal of 294, 1/294 is 42 digits long. SO WHAT? Wikipedia says that there is number with the same length in base ten? Seems incredible.<br /><br />294 is the area of a Heronian triangle, with all integer sides (can you find it?)<br /><hr /><b>The 295th Day of the Year</b> <br />295 may be interesting only because it seems to be the least interesting day number of the year. (Willing to be contradicted, send your comments) [<i>Here are several of the best I received from David Brooks: 295 can be partitioned in 6486674127079088 ways. 295 is a 31-gonal number.</i>]<br />Prime Curios has it as another of those numbers where the sum of the products of the number times each digit is a prime, 2(295) + 9(295) + 5(295)is primes. <br /><br />And Derek Orr pointed out that "295 is the second proposed Lychrel number." A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been yet proved to exist, but many, including 295, are suspected on heuristic and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of Cheryl, his girlfriend's first name. (Who else thinks he probably mis-spelled her name and when she called him on it, he came up with the idea of a "rough anagram"? )<br /><br />Lagrange's Thm says all positive integers can be written with no more than four positive squares. 295 is the 48th of the ones that require the full four, 17^2 + 2^2 + 1^2 + 1^2 works. There are sometimes adjacent numbers that require four (111,112) was the first, but there can never be a triplet. <br /><br />295 is the sum of four consecutive tetrahedral numbers (the sum of the firs n triangular numbers) 120 + 844 + 56 + 35 <br /><br />295 = 32^2 - 25^2 =148^2 -147^2 <br /><hr /><b>The 296th Day of the Year</b> <br />296 is the number of partitions of 30 with distinct parts. (Even very young students can enjoy exploring the number of partitions of integers, and the difference in the number when the parts must be distinct. The idea can be explored for very young students with number rods, etc)<br /><br />A cube with an 8x8 checker board on each face has a total of 296 lattice points (where the squares meet) <br /><br />Derek Orr point out that 296^4 is a ten digit number, but only uses the digits 2,3456 The somewhat famous "look and say" sequence in math, 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, (the second term is 11 because the previous term has One, one; etc) has 296 digits in the 18th term. <br /><br />Derek Orr points out that 296^4 is a ten-digit number that only uses five consecutive digits, 3, 4, 5, 6, and 7. It ends in 3456. If you add one, everything the same except it ends in 3457, AND, it's a prime. 296 is the 44th year day for which n^4 + 1 is prime. <br /><br />Both 296 and 297 are of the form p^3 q where p and q are primes. The only other consecutive pair of year days is 135, 136. Because of this they also each have the same number of divisors (8)<br /><br />Because 296 = 2^3 x 37, the factors of two lead to several expressions of 296 as difference of squares, 75^2 - 73^2 = 296 = 39^2 - 35^2. And as 296 / 4 = 74 = 5^2 + 7^2, 296 = 10^2 + 14^2. (Students might be curious why nothing like that happens with 297.) <br />296 is also the difference of two cubes, 8^3 - 6^3.<br /><br />296 is called a refactorable number, it is divisible by the number of divisors it has (8).<br /><hr /><b>The 297th Day of the Year </b><br />297<sup>2</sup> = 88209 and 88+209 = 297. <br />Such numbers are called Kaprekar numbers. This is the last day of the year that is a Kaprekar number, named for D R Kaprekar who wrote about them. Every Kaprekar number has a ten-pal (my cutesy variant of a pen-pal) if a n digit number is a Kaprekar number, k, then 10^n - k will also be one. For 297, the ten-pal will be 10^3 - 297 = 703. 703^2= 494209 and 494+209=703.<br /><br />297 is another 6n+9 number, so it can be written as the difference of two squares, 51^2 - 48^2, and of course, like all odd numbers, it is the difference of two consecutive squares, 149^2 - 148^2.<br /><br />And this one from * Jim Wilder @wilderlab 297<sup>3</sup>=26,198,073 and 26+198+073=297. Not a Kaprekar number, but even "Wilder". 297 is only the sixth Kaprekar number. There is only one known number that preserves the relationship in 2nd, 3rd, and fourth power, it is 45.<br /><br />Both 296 and 297 are of the form p^3 q where p and q are primes. The only other consecutive pair of year days is 135, 136. Because of this they also each have the same number of divisors (8). <br /><br />Prime Curios points out an interesting relation for 297. If you write a 43 digit number with twenty-nine 7's followed by seven 29's, it's prime. If you reverse that and write seven 29's followed by twenty-nine 7's, still prime.<br /><br />One from my notes, 29 + 79 + 97 + 92 = 297 (palindrome with only the three digits of n. <br /><hr /><b>The 298th Day of the Year</b> <br />If you multiply 298 by (298 + 3) you get a palindromic number, 89,698. Can every number be similarly adjusted to make a palindrome? And this one is not just a Palindrome, it's a strobogrammatic one, rotate it 180 degrees and you get another palindrome, 86968. (Some restrict the term stobogram only to numbers that recreate themselves after rotation, and prefer ambigram for the ones that rotate to make a different number.)<br /><br />298 = \( {12 \choose 1} + {12 \choose 2} + {12 \choose 3} \) This is related to the <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/eggold.pdf" target="_blank">Egg Drop numbers</a><br /><br />6 x 298 +/- 1 are twin primes. The 55th number of the year for which this is true. <br /><br />298 and 299 have the same number of divisors. All semi-primes(product of two primes) have four divisors. 298 is the 45th number that has the same number of divisors as it's successor. <br /><br /><hr /><b>The 299th Day of the Year</b> <br />If a cubic cake was cut with 12 straight cuts, it can produce a maximum of 299 pieces.... a good day to "let 'em eat cake."<br /><br />An annulus (2-D donut) would require 23 cuts to produce the same number . There are 299 semiprimes less than 1000 which are products of two primes. <br /><br />and 299 is a semi-prime, but with both factors having the same length (number of digits). Only four year days left share this property. <br /><br />299 = 150^2 - 149^2. <br /><br />299 = 99 + 1 + 99 + 1 + 99 <br /><br />299 is the long leg of a Pythagorean triangle. (180, 299, 349)<br /><br /><hr /><b>The 300th Day of the Year </b><br />300 is a triangular number, the sum of the integers from 1 to 24.<br /><br />300 is also the sum of a pair of twin primes (149 + 151). And the sum of ten consecutive primes, 300 = 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47. <br /><br />300 = 2 x 5 x 3 x 5 x 2, palindrome product of primes.<br /><br />300 is the largest natural number that is Not the sum of a prime and a 3-almost primes (product of three primes, not necessarily distinct). 7+2 x 3 x 5 = 37 can be so expressed. Students might find other numbers that are no expressible as such.*Prime Curios <br /><br />300 is is a palindrome in three consecutive bases, base 7(606), base 8 (454), and base 9 (363). And in Roman Numerals it is a repdigit, CCC<br /><br />300 can be expressed as the difference of two squares in three different ways, 76^2 - 74^2 = 28^2 - 22^2 = 20^2 - 10^2 = 300 <br /><br />Oh Dear! 300 is not a Happy Number. Iterations of the sum of the squares of the digits fall into the eight- cycyle 3 --- 9 --- 81 --- 65 --- 61 --- (37 --- 58 --- 89 --- 145 --- 51 ---26 --- 40 --- 16 ----37)<br /><br />300 is the 49th Day of the Year for which n^2 + 1 is prime. <br /><br />The Fibonacci sequence Modula 50, has a period length of 300. As an example for a smaller number, mod 2, the numbers 1, 1, 2, 3, 5, 8, 13 Mod 2 have residues 1, 1, 0, 1, 1, 0, 1, for a repeating pattern of 3. <br /><br /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-43760500084028339122020-08-01T21:01:00.031-07:002021-04-17T11:22:04.256-07:00Number Facts for every Year Date, 241 - 270<b>The 241st Day of the Year</b><br />The 241st day of the year; 241 is the larger of a pair of twin primes. The larger of a pair of twin primes is always one more than a multiple of six; the smaller is always one less than a multiple of six. <br /><br />2+4+1 is prime. 241 is the 53rd prime. (53 is also prime) *Derek Orr<br /><br />241 is also The smallest prime <i>p</i> such that <i>p</i> plus the reversal of <i>p</i> equals a palindromic prime. 241 + 142 = 383; which is a prime palindrome. <br /><br />And it is the largest known prime p such that the reversal of (p! + p) is prime. (241! + 241 ends with a string of fifty-five zeros, and then 241 : 980360372638941007038951797078339359751464353463061342202811 188548638347461066010066193275864531994024640834549254693776 854464608509281547718518965382728677985343589672835884994580 815417004715718468026937051493675623385569404900262441027874 255428340399091926993707625233667755768320823071062785275404 107485450075779940944580451919726756974354635829128751944137 27644867102380111026020691554782580923999494640500736 0000000000000000000000000000000000000000000000000000000241 and if you write the reversal of that, it's prime. <br /><br />241 is the smallest non-palindromic prime which can be expressed as sum of a number and its reverse, i.e., 241 = 170 + 071. *Prime Curios. <br /><br />241 is the smallest prime p such that p^7 can be written as the sum of seven consecutive priems. *Prime Curios<br /><br />241 can be written as the sum of three double digit emirps, 71 + 73 + 97. There is no larger number for which this is true. *Prime Curios<br /><br />241 = 121^2 - 120^2.<br /><br />241 = 15^2 + 4^2 which means 241 is a Pythagorean Prime.<br /><br />241 is a palindrome in duodecimal, base 12 (181) and a repdigit in base 15(111)<br /><br />P. Honaker at Prime Curios points out that the sequence of primes formed by n!+239 begins, 241, 263, 359... Maybe I mis-searched but I did not find this sequence in OEIS. Seems like a good computer programming project for students, pick a prime and find primes of the form n! + p <br />John Cook posted that "if k is relatively prime to b, there is a multiple of k whose base b representation contains all ones. If I understand that then since 241 is prime, it is relatively prime to ten. Can you find the base ten multiple of k that has all ones for its digits? Be the first to share the answer with me and get imortality by being listed here. I think it must be a very large multiple of 241. <hr />, <b>The 242nd Day of the Year</b><br />242 has six divisors...but 243, 244, and 245 also each has six divisors. 242 is the smallest integer to begin a run of four consecutive integers all of which have the same number of divisors. (What is the smallest integer that begins a run of three consecutive integers with an equal number of divisors?)<br /><br />242 is not only a palindrome in base ten, it is also a palindrome (and repdigit) in base 3, \( 22222_3\) and base 7, \( 464_7 \). (What palindrome in base ten is also a palindrome in the most other bases 2-9?)<br /><br />242 = 2 x 11^2, it is divisible by 2, 11, 22, 121 all of which are also palindromes.<br /><br />242 is the smallest number that is between a prime and a fifth power of a prime.*Prime Curios <br /><br />242 is one more than the larger of a twin prime pair, and 6 x 242 is between a prime pair. <br /><br />242 is a multiple of 11, and if you sum the digits in base 100 (ie, 2 + 42) it is also divisible by 11. <br /><div><br />242 us the nth prime, + n for n = 45. <br /><hr /><b>The 243rd Day of the Year</b> <br />243 is the largest three digit number that can be expressed as a fifth power (3<sup>5</sup>). It is the smallest p^q where p and q are twin primes.<br /><br />243 is also the sum of five consecutive prime numbers (41 + 43 + 47 + 53 + 59).<br /><br />243 is the sum of palindrome primes, 11+101+131 *Prime Curios<br /><br />243 is the sum of 4# + 3# + 1# + 0#, where n# is the product of the first n primes and 0#=1, 243 = 210 + 30 + 2 + 1 *Prime Curios<br /><br />Venus' day is 243 Earth days. *Derek Orr<br /><br />243 is a Harshad number, divisible by the sum of it's digits, and every permutation of its digits is also. <br /><br />243, like all odd numbers, is the difference in consecutive squares, 122^2-121^2 = 243, because it is 3 mod<sub>6</sub> it is also the difference of two squares of integers that differ by 3, 42^2 - 39^2. <br /><br />243 is a palindrome in base 8 (363)<br /><br />3^3 + 6^3 = 243</div><div><br /></div><div>On April 14, 2014, Almost exactly a year after Yitang Zhang announced a proof (see April 17) that there are infinitely many pairs of prime numbers which differ by 70 million or less Terrance Tao's online group attack on the problem reduced the number to 243. Zhang's proof is the first to establish the existence of a finite bound for prime gaps, resolving a weak form of the twin prime conjecture. </div><div><hr /><b>The 244th Day of the Year</b><br />244 is the smallest number (besides 2) that can be written as the sum of 2 squares (10^2 + 12^2) or the sum of two 5th powers(5^3 + 1^3) <a href="http://www2.stetson.edu/~efriedma/numbers.html" target="_blank">*What's Special about this number</a> It is the second smallest number where both squares are squares of numbers with more than one prime factor.<br /><br />244 is (the second) anti-perfect. The proper divisors are 1, 2, 4, 61, and 122, & adding their reversal is 1 + 2 + 4 + 16 + 221 = 244. *Jim Wilder @wilderlab (<i>244 is the smallest multi-digit anti-perfect number; There is one more year day which is anti-perfect</i>.... don't just sit there, go find it!)<br /><br />244 is also the sum of three cubes, \( 244 = 1^3 + 3^3 + 6^3 \)<br /><br />244 is "power friendly" with 136. \(244 =1 ^3 + 3^3 + 6^3\) and \(136 = 2^3 + 4^3 + 4^3\)<div><br />244 = 2^2 x 61, and because it is divisible by four, it is the difference of two squares, 62^2 - 60^2. <br /><br />244 is a palindreome in base 3 (100001) and base 11 (202) <br /><br />244 is the 7th number in the "double and reverse" sequence, 1, 2, 4, 8, 61, 221, 244, ... *Wikipedia <br /><hr /><b>The 245th Day of the Year</b><br />245 is the fifth StellaOctangula number. The sum of the 5th octahedral number (85) and eight of the fourth tetrahedral numbers (20). 245 =85 + 8 (20)<br /><br />245 is also the sum of three consecutive squares, \(245 = 8^2 + 9^2 + 10^2 \)<br /><br />There are 245 odd entries in the first 33 rows of the Arithmetic Triangle.<br /><br />245 is also the 46th prime plus 46, 199+46=245 (is there a mathematical significance for these numbers, or just a nice curiosity? Serious question.)<br /><br />The third largest known pair of "cousin primes, p and p+4) is given by the product of all odd numbers up to 245 +/- 2, they each have 241 digits<br /><br />245 is the aliquot sum (sum of the divisors less than n) of any of these numbers: 723, 1195, 2563, 3859, 9259, 10123, 12283, 14659 and 14803.<br /><br />The smallewst base that makes 245 base b a palindrome is base 34, where it is the repdigit 77. After base 36= 26 + 9 there are no letters left, how do thewy indicate a number in bases greater than 37 that is 37, 38, etc.<br /><br />245, since it ends in 5 and is bigger than 25, is the difference of two squares of integers that differ by 5.. 245 = 27^2 - 22^2. Like all odd numbers, it is the difference of consecutive squres, 123^2 - 122^2.<br /><hr /><b>The 246th Day of the Year</b><br />246 is a sphenic (wedge) composite since it is the product of three distinct prime factors, 246 = 2x3x41. (what would be the next sphenic number?)<br /><br />246 is also equal to the sum 9C2 + 9C4 + 9C6 (9 choose 2,4,6)<br /><br />246 = 233 + 13 (13th Fibonacci number plus 13) *Derek Orr@Derektionary<br /><br />246 is the smallest number whose complete factorization contains the first four digits (and no others) 246 = 2*3*41<br /><br />Algebra Fact @AlgebraFact points out that: "Tao, et all, have proved that there are infinitely many primes <b>246</b> apart." On the way to proving, hopefully, that there are an infinity of twin primes. <br /><br />246 is an untouchable number, a number that can not be formed from the proper divisors of any number. Paul Erdos proved there are an infinite number of them. The early ones are 2, 5, 62, 88, 96... There are only 29 untouchable year dates, but very unevenly divided by the century groups. 5 below 100, 5 more before 200, then 12 between 200 and 300, and 8 between 300 and 400. <br /><br />246 is a palindrome in base 5(1441) and base 9(303) <br /><br />246 can be expressed as the sum of three (not distinct) squares. 14^2 + 5^2 + 5*2 <br /><br />The "aliquot sequence) of a number is the chain of results following from iterating the sum of the aliquot factros. The Chain for 246 is 258, 270, 450, 759, 393, 135, 105, 87, 33, 15, 9, 4, 3, 1, 0. *Wikipedia <br /><br />246 is the average of two primes br /><br /><hr /><b>The 247th Day of the Year</b> <br />247 is the smallest number which can be expressed as the difference between two integers such that together, they contain all digits 0-9. (50123 - 49876)<br /><br />The digits of 247 sum to its smallest prime factor (247 = 13 x 19 and 2 + 4 + 7 = 13) *Prime Curios 247 = 13 x 19. The two factors are prime when concatenated in either order.(<i>How many of the composite days of the year sum to one of their prime factors?</i>)<br /><br />The digits of 247 add up to 13, which divides 247 making it a Harshad (Joy-giver) number. It is the smallest Hareshad Nummber with a digit sum of 13.<br /><br />The mathematician and philosopher Alex Bellos suggested in 2014 that a candidate for the lowest uninteresting number would be 247 because it was, at the time, "the lowest number not to have its own page on English Wikipedia". *Wikipedia (OF course that fact makes it interesting, proving that there are no uninteresting numbers. <br /><br />247 is the difference of consecutive squares, 144^2 - 143^2 , and 16^2 - 3^2 also. <br /><br />247 is the 13th Pentagonal number ( n x (3n-1)/2) . The average of the first n pentagonal numbers, is the nth Triangular number. Like all pentagonal numbers, it is the sum of the n consecutive numbers starting with n. so the 13th pentagonal number is 13 + 14 + 15 + ... 25 = 247<br /><br />And the number 66 can be partitioned into squares in 247 different ways. <br /><hr /><b>The 248th Day of the Year</b><br />248 is the smallest number (above 1) for which the arithmetic, geometric, and harmonic means of φ(n)(Tne umber of positive integers less than n that are relatively prime to it, 120) and σ(n) (the sum of positive integers less than n that are factors of n, 480) are all integers.<br /><br />248 = 2<sup>8</sup> - 2<sup>3</sup> , which can also be expressed as 2^n - n<br /><br />248 is the only number such that its unique prime factorization is of the form \(p^{p+1}[p^{p+3}-1]\) *Prime Curios<br /><br />248 is an untouchable number, there is no number n from which a subset of its divisors sum to 248.<br /><br />248 is also a refactorable number since it is divisible by the count of its own divisors (8).<br /><br />Because it is divisible by four and by eight, it is the difference of two squares as (63^2 - 61^2) and (33^2 - 29^2), <br /><br /><hr /><b>The 249th Day of the Year</b><br />The 249th day of the year. 249 is the index of a Woodall prime. A Woodall number is a number of the form W(n) = n(2<sup>n</sup>)-1. The first few are 1, 7, 23, 63, 159, 383, ... (Sloane's A003261). W(249) is prime. [Proof left to the reader, ;-} ] W(2)=7; W(3)=23 and W(6)=383 are all prime. What's the index of the next prime Woodall number? ( <i>named after H. J. Woodall who studied them in 1917</i>)<br /><br />249 = (3!)<sup>3</sup> + (2!)<sup>5</sup> + (1!)<sup>7</sup> (consecutive odd powers of consecutive factorials) *Derek Orr <br /><br />and Jim Wilder @wilderlab sent 249 <sup>3</sup>= 15,438,<b>249</b> Check 249<sup>2n-1</sup> and be surprised, then find out why.<br /><br />Because 249 is equivalent to \(3 mod_6\) it is the difference of two squares of integers three apart, \(43^2 = 40^2 = 249\). And also the difference of two consecutive squares, 125^2 - 124^2. <br /><br />249 = 10^2 + 10^2 + 7^2, and also 14^2 + 7^2 + 2^2<br /><br />Thee first 34 rows of the arithmetic triangle has 259 odd numbers. (How many even?)<br /><br />249 is two less than a prime, and 249^2 is also two less than a prime. And its cube, which is the 12th Fermat Prime; but alas the fourth power has a factor of 3. </div><div><br /></div><div>249 is the last year day which can be written as the sum of prime numbers using only the digits 1-5 exactly once in the digits of the prime in only one way: 249 = 241 + 5 + 3: The largest year day that can be written in such a manner in more than one way is a permutation of 249, 294 = 43 + 251 = 53 + 241.</div><br />168 and 249 have an interesting relationship, the sum of their digits are equal, and the sum of the squares of their digits are equal. \( 1 + 6 + 8 = 2 + 4 + 9 = 15\) and \(1^2 + 6^2 + 8^2 = 101 = 2^2 + 4^2 + 9^2\) <hr /><b>The 250th Day of the Year</b><br />250 is the smallest number expressible as the sum of two positive cubes, which is also expressible as the sum of two (unique) positive squares in more than one way. ( on Feb. 16, 1745 Euler wrote to Goldbach and shows that numbers represented in two different ways as a sum of two squares must be composite; hence 250 is composite. )<br />5<sup>3</sup>+ 5 <sup>3</sup> = 250<br />15<sup>2</sup>+ 5 <sup>2</sup> = 250 ; 13 <sup>2</sup> + 9<sup>2</sup> = 250<br />250 is only the 18th number that is the sum of two positive cubes <br /><br />and 250 = (3!)<sup>3</sup> + (2!)^<sup>5</sup> + (1!)<sup>7</sup>+ (0!)<sup>9</sup> *Derek Orr<br /><br />Strange that I would learn this from a Brit, but "Length of a baseball pitch, pitcher to batter (18.44 m) is 250 x Diameter of regulation baseball (73.7 mm) *http://www.isthatabignumber.com <i>He used the term for the "pitch" (</i><i>pitching </i><i>or bowling area) in cricket, and btw, the cricket pitch is 1 chain, or 22 yards long, and that's 20.12 meters (but the batting (popping) crease is almost 1 1/4 meters in front of the stumps, so the pitching distances are very similar. </i> <br /><br />250 is the smallest multidigit prime where the sum of the squares of its prime factors is the same as the sum of the squares of its digits. *Prime Curios <br /><br />250 = 1^2 + 2^2 + 7^2 + 14^2, the sum of the squares of its divisors. <br /><br />250 is the 49th prime - 1. Its square is also one less than a prime<br /><hr /><b>The 251st Day of the Year</b><br />The four consecutive primes 251-257-263-269 all have prime gaps of 6. There are no other consecutive prime gaps for the rest of the year. There have been four primes that had the same prime gap before and after them. There are gaps of six on each side of 53, 157, and 173; and a gap of two on both sides of 5. <br />251 is a prime number that is also the sum of three consecutive primes (79 + 83 + 89) and of seven consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47). In addition, it is the smallest integer that can be the sum of three cubes in two different ways. 251 =2<sup>3</sup>+ 3<sup>3</sup>+6<sup>3</sup> = 1<sup>3</sup>+5<sup>3</sup>+5<sup>3</sup> <br />and there are 251 primes less than 1600.*@MathYearRound <br /><br />and wow: The 251st Fibonacci number (12776523572924732586037033894655031898659556447352249) has a sum of digits equal to 251. *jim wilder @wilderlab <br /><br />251 is the 49th Prime, and the largest prime day year that is the concatenation of two squares. <br /><br />251 = 2^3 + 3^5 <br /><br />All the prime numbers up to 251, without repetition, appear in Benjamin Franklin's original 16-by-16 square. Prime Curios<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-xYQy5sG8KqM/XxnygzJVFzI/AAAAAAAAM6w/zytCWrKw3tU-hy7rs5BXOpnHIijLEbl1ACLcBGAsYHQ/s1600/16x16%2Bmagic%2Bsquare%2BFranklin.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="458" data-original-width="522" height="281" src="https://4.bp.blogspot.com/-xYQy5sG8KqM/XxnygzJVFzI/AAAAAAAAM6w/zytCWrKw3tU-hy7rs5BXOpnHIijLEbl1ACLcBGAsYHQ/s320/16x16%2Bmagic%2Bsquare%2BFranklin.jpg" width="320" /></a></div><br />I found the <a href="http://www.math.wichita.edu/~richardson/franklin.html">image here</a>, with lots of notes on additional magic components. <br /><br />251 is the smallest number that is the sum of three cubes in two ways, 1^3 + 5^3 + 5^3 = 2^3 + 3^3 + 6^3 *Prime Curios <br /><br />In 1641 Mersenne stated that \(2^251 - 1\) is composite, without proof. Prime Curios (The smallest factor is 503.) <br /><br />251 can be formed by the difference of two five digit numbers that share no common digit, and thus contain all the digits from 0-9.<br />40126 - 39875 <br /><br />251 is a palindrome in base 8 (373)<br /><br />251 is the difference of two consecutive squares, 126^2 - 125^2 <br /><br />251 is a Sophie Germain prime since 2*251+1 is also prime. <br /><hr /><b>The 252nd Day of the Year</b><br />The 252nd day of the year; 252 is the smallest number which is the product of two distinct numbers that are reverses of each other: 252 = 12*21 *Number Gossip <br />(and the next would be?)<br /><br />252 is a palindrome in base ten, and also in base five 2002<sub>5</sub> (How many three digit numbers are (non-trivial) palindromes in base ten and one other base less than ten)<br /><br />If you flip a coin 10 times in a row, there are exactly 252 ways in which it can turn out that you get exactly 5 heads and 5 tails. That is, \( 252 = \binom{10}{5} = \frac{10*9*8*7*6}{5*4*3*2*1} \) <br /><br />252 = 2^2 x 3^2 x 7. Because it is divisible by four, it is the difference of two squares, 64^2 - 62^2 , but it is also expressible as 24^2 - 18^2, and 16^2 - 2^2.<br /><br />252 is divisible by the number of its divisors (18) and is called a refactorable number. <br /><br />252 is the largest year day that is the sum of the squares of the numbers in a row of Pascal's triangle, the squares of the fifth row are \( 1^2 + 5^2 + 10^2 + 10^2 + 5^2 + 1^1 = 252 \) </div><div><br />252 is also a practical number, since every integer smaller than 252 can be formed by adding a subset of the divisors of 252. <br /><br />252 can be expressed as the sum of prime numbers using only the digits 1-9 once each. 2 + 41 + 59 + 83 + 67. (You can do the same with 225) Clever students will see an alternate way to do 252 manipulating the one above.)</div><div><br />252 is the sum of four cubes (1^3 + 2^3 + 3^3 + 6^3) It can also be expressed as (1^3 + 2^3)(1^3 + 3^3) <br /><hr /><b>The 253rd Day of the Year</b><br />253 is the 22nd triangular number, and thus the number of combinations of 23 things taken two at a time. More unusual for a triangular number, it is also the 9th centered heptagonal number. It seems there there are only five known numbers that are both triangular and centered heptagonal. (I found it interesting that if you find the digital roots of the centered hexagonal numbers, the sequence of the digital roots has period 9: repeat[1, 8, 4, 7, 8, 7, 4, 8, 1] (the period is a palindrome))<br />Although there are 116 semiprimes in the year, 253 is the last semiprime year day that is a triangular number. 253 = 11 x 23.</div><div><br /></div><div><br />253 can be written as the sum of consecutive natural number in three different ways, including 1+2+.... + 22, 18 + 19+... + `28 , and 126+127,<br /><br />253 is a palindrome in base 12(191)<br /><br />2<sup>5</sup>-3 is prime. Derek Orr pointed out that 2+5<sup>3</sup> is also prime, in fact, it’s a Mersenne prime, M<sub>7</sub> <br /><sub>(Ok, there has to be lots more interesting things about 253, it has all Fibonacci prime digits, so there must be cool stuff out there I'm missing, HELP!?</sub><br /><hr /><b>The 254th day of the Year</b><br />254 is the maximum number of pieces a flat pizza could be cut into with n straight lines.... find n. (for help, see bottom of this post, a good quadratic problem)<br /><br />254 is the average of consecutive primes,<br /><br />and 254 = 2^8 - 2^1<br /><br />Probability Fact @ProbFact points out that: Odds of drawing a straight in a 5-card hand: 254 to 1.<br /><br />254 is a day so with so few interesting facts, that as a teacher, you always want it to be on a weekend.! <br /><hr /><b>The 255th Day of the Year </b><br />255= 2<sup>8</sup>-2^0 and is the fourth Mersenne number that is not a prime. However it is the product of three distinct Fermat Primes, 3*5*17, and therefore the regular 255-gon is constructible with straightedge and compass. *HT to Don S. McDonald @McDONewt who also pointed out that the next two numbers, 256 and 257 are also constructible since one is a power of two, and the other is a Fermat Prime. <br /><br />255 is a also a repdigit in base 2 (11111111) in base 4 (3333), and in base 16 (FF). (<i>What is the next number that is a repdigit in base two and base 4?) </i><br /><i><br /></i><i>John D Cook has a <a href="http://www.johndcook.com/blog/2011/09/09/five-interesting-things-about-mersenne-primes/" target="_blank">nice overview of Mersene Numbers and Mersene Primes</a></i><br /><br /><a href="https://2.bp.blogspot.com/-t8fntNA8mfc/V82QyX3RaqI/AAAAAAAAIcg/_PWCLZ1FTfUTH4_L3fMa1MrJwRV468vVACLcB/s1600/pacman.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="182" src="https://2.bp.blogspot.com/-t8fntNA8mfc/V82QyX3RaqI/AAAAAAAAIcg/_PWCLZ1FTfUTH4_L3fMa1MrJwRV468vVACLcB/s200/pacman.png" width="200" /></a> 255 is the number of levels on the Pac-Man arcade machine prior to the "kill screen" rendering game over... why 255? (Computer people know )*Jim Wilder@wilderlab<br /><br /> In the 3n+1 or <a href="https://en.wikipedia.org/wiki/Collatz_conjecture" target="_blank">Collatz problem</a>, the sequence for n = 255 reaches higher than any other year day, to the value of 19,682. The previous high value was at 27, when it reached 9232.(The Collatz Problem seems to have deep relationships in number theory, but no one seems to know how to trace them down. A Medus- like problem with a thousand heads that each distract us from their common purpose.<br /><br />255 is the difference of consecutive squares, 128^2 - 127^2 . It is also 28^2 - 23^2, and of 44^2 - 41^2. and 16^2 -1^2.<br /><br /><hr /><b>The 256th Day of the Year</b><br />256 is the smallest composite to composite power,4<sup>4</sup>.<br />A student note for fourth powers, They are always the sum of two (I think consecutive) triangular numbers, 2^4 = 6+10 (t(3) + t(4)); 3^4 = 36 + 45 (t(8) + t(9)); 4^4 = 120 + 136 (t(15) + t(16)). There is a pattern in the index of the larger triangular number of each pair.</div><div><br />256 is also 2^8 and 16^2 (2^8 let's me show off a fancy word for eighth powers, <b>zinzizinzizinzic</b> which essentially means the square of a squared square.<a href="https://pballew.blogspot.com/2020/05/volume-t-z.html"> More about the term and it's history here</a>.<br />perfect powers like 2^8 are pretty rare, with 256 being only the 23rd of the year. There will be only four more until years end. (Yes, of course you want to find them!)<br /><br />Paul Erdos conjectured that no power of 2 is the sum of distinct powers of three.<br /><br />from jim wilder @ wilderlab √256 = 2 • 5 + 6<br /><br />The sum of the cubes of the first 256 odd numbers is a perfect number. \( \sum\limits_{i=0}^{255} (2i+1)^3 = 8589869056\) the 6th perfect number. (all perfect numbers (<i>except 6</i>) are the sum of the cubes of first 2<sup>n</sup> odd cubes for some (<i>but not all</i>) n) (so \( 28 = 1^3 + 3^3\) and \( 496=1^3 + 3^3 + 5^3 + 7^3\) ).<br /><br />256 is the middle number in a run of three successive numbers which are all constructible regular n-gons. 255= 3*5*17, is the product of distinct Fermat Primes, 256=2<sup>8</sup> and is a power of two, and 257 is a Fermat Prime. *HT to Don S. McDonald @McDONewt <br /><br />With so many factors of two, 256 is expressible as the difference of two squares in several ways. 65^2 - 63^2, 34^2 - 30^2, and 16^2 - 0^2<br /><br />256 is a near perfect number, the sum of its aliquot divisors is 255. All powers of two are near perfect (aliquot sum is n-1) but it seems there are no known number(s) which is abundant by one.OEIS <br /><br />256 is one less than a prime (257). It is the eighth power of two, and four of them are one less than a prime, four are not.... but in the long run??? <br /><hr /><b>The 257th Day of the Year</b><br />The four consecutive primes 251-257-263-269 all have prime gaps of 6. There are no other consecutive prime gaps for the rest of the year. There have been four primes that had the same prime gap before and after them. There are gaps of six on each side of 53, 157, and 173; and a gap of two on both sides of 5. </div><div> <br /><span style="font-family: "courier new" , "courier" , monospace;">257 is a prime number of the form 2</span><sup style="font-family: "courier new", courier, monospace;">2<sup>3</sup></sup><span style="font-family: "courier new" , "courier" , monospace;">+1 and therefore a Fermat prime. It is currently the second largest known Fermat prime. It is the only known Fermat Prime that is not part of a Twin Prime pair.</span></div><div><span style="font-family: courier new, courier, monospace;"><br /></span><span style="font-family: inherit;"><span br="" curios="" prime="" style="font-family: "courier new" , "courier" , monospace;"><br /> 257 is the third consecutive number (255,256,257) for which the regular n-gon is constructible with straightedge and compass. The 255th day of the year; 255= 2<sup>8</sup>-1 is the product of three distinct Fermat Primes, 3*5*17, 256=2<sup>8</sup> is a power of two, and 257 is a Fermat Prime *HT to Don S. McDonald @McDONewt (Goldbach used the fact that all Fermat Numbers are 2 + product of all smaller Fermat Primes to prove that no two Fermat Numbers share a common prime divisor)<br /><br /> At one time, it was announced that \(2^{257} -1) was prime also, and thus a new perfect number had been found. On March 27,1936 The Associated Press released a story that a new 155 digit perfect number had been found by Dr. S. I. Krieger of Chicago. The number was \(2^{256}(2^{257} - 1)\) by proving the \(2^{257} -1\) was prime. This was shocking since D. H. Lehmer and M. Kraitcik had announced that the number was composite in 1922. Unfortunately, their method did not include giving a factor of the number. The perfection of the number was doubted by most mathematicians, but the actual factoring to prove it was composite didn't happen until 1952 when the SWAC confirmed it was composite by finding a proper divisor. *Beiler, Recreations in the Theory of Numbers. </span></span></div><div><span style="font-family: inherit;"><span br="" curios="" prime="" style="font-family: "courier new" , "courier" , monospace;"><br /></span></span></div><div><span style="font-family: inherit;"><span br="" curios="" prime="" style="font-family: "courier new" , "courier" , monospace;"> 257 = 4<sup>4</sup> + 1 It is the largest known prime of the form n<sup>n</sup> + 1. *Prime Curios 2^2 + 1 is also prime, are there more? <br /><br /><br /><br /> More than 90% of all positive integers are composite numbers that have a lowest prime factor of 257 or less. (Would this be equally true for nearby primes like 251 or 263? It seems like as the prime grows in value, the Pctg would slowly diminish. Yes? No? Teach me.)<br /><br /> 2<sup>257</sup> - 1 is the largest number in Mersenne's list of primes in the preface to his Cogitata Physica-Mathematica (1644), it later turned out to be Composite. * Dan Garbowitz @DGoneseventh<br /><br /> Ones and zeros, 257 written in different bases, 100000001<sub>2</sub>, 10001<sub>4</sub> 101<sub>16</sub> (Students, Why is eight not in this list?) <br /><br />257 = 2^8 + 2^0 <br /><br />Prime Curios points out that 257 is the largest of 15 consecutive primes of the form 2T + 1 where T is the triangular numbers. 1, 3, 6, ... <br /><br />257 is a factor of 1068349. This division problem includes all the digits from 0 - 9 once each. <br /><br />257 is a palindrome in base 2 with LOTS of zeros, (100000001) and in base 4( 10001) and yes, in base 16 (101) <br /><br />There is(was?) a Pac-Man themed restaurant called Level 257 located in Schaumburg, Illinois. It is in reference to the kill screen reached in Level 256 in the Pac-Man arcade game. I checked online and in the midst of the Pandemic they are open daily with games, and shortlane bowling (No more gutter balls for this boy!).</span></span></div><div><span style="font-family: courier new, courier, monospace;"><br /></span></div><div><span style="font-family: courier new, courier, monospace;">257 = 5^3 x 2^2 - 3^5 Is it possible to express all primes as A x B +/- C where A, B, and C are each a power of bases 2, 3, and 5 (all distinct)<br /></span><b></b><br /><hr /><b>The 258th Day of the Year</b> <br />258 is a sphenic(wedge) number (the product of three distinct prime factors..258 = 2·3·43) it is also the sum of four consecutive primes 258 = 59 + 61 + 67 + 71 <br /><br />(Jim Wilder@Wilderlab pointed out that 2,5,&8 are the numbers in the center column of a phone or calculator.) Jim's comment reminded me of a math type phone joke I saw at Wolfram Mathworld: "I'm sorry, the number you have dialed is an imaginary number. Please rotate by 90<sup>o</sup> and try again." Taking this joke one step further gives the "identity" \( 8*i = \infty \) And that reminds me of<a href="http://blog.mindresearch.org/blog/imaginaries-at-play-funny-math-cartoon" target="_blank"> this cartoon</a> at Mind Research Institute.<br /><br /> The <a href="http://www.archimedes-lab.org/numbers/Num1_69.html" target="_blank">Number Zoo</a> gives a Magic square using 16 consecutive primes, with a constant of 258 <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-1Cd3h1F4VeA/Ve7q4r4ZQKI/AAAAAAAAG94/eKxocvqvIFQ/s1600/prime%2Bmagic%2Bsquare%2B4x4.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-1Cd3h1F4VeA/Ve7q4r4ZQKI/AAAAAAAAG94/eKxocvqvIFQ/s1600/prime%2Bmagic%2Bsquare%2B4x4.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">*Prime Curios points out that <span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">258 is the minimum magic sum of a magic square utilizing sixteen consecutive </span><a class="glossary" href="https://primes.utm.edu/glossary/xpage/Prime.html" style="background-color: white; border: 1px dashed rgba(0, 51, 0, 0.25); box-sizing: border-box; color: #003300; cursor: pointer; font-family: Roboto, sans-serif; font-size: 16px; padding: 0px 2px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s;" title="glossary">primes</a><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">. First found by Allan W. Williams, Jr., of Washington, D.C. <</span></div><br /><br />Prime Curios also states that if you take all six permutations of 258, put them in order, and make the 18 digit concatenation of them, it is the composite number between a prime pair. Can you find others like this?<br /><br />258 is the sum of four consecutive Prime numbers, 59 + 81 + 67 +71 *Wikipedia<br /><br />258 = 6^3 + 6^2 + 6, or 1110 in base six. Tomorrow that will be 1111.<br /><br />258 is the sum of two consecutive primes 127 + 131. <br /><br />and 258 = 2^8 + 2^1. <br /><hr /><b>The 259th Day of the Year</b><br />259 expressed in base six is a repunit, 1111 (6<sup>3</sup>+6<sup>2</sup>+ 6<sup>1</sup>+6<sup>0</sup>= 216+36+6+1=259). And also in base 36, which we all use often, its 77. <br /><br />259 can be expressed as the sum of four cubes in two different ways, 259 = 1<sup>3</sup> + 2<sup>3</sup> + 5<sup>3</sup> + 5<sup>3</sup>= 2<sup>3</sup> + 2<sup>3</sup> + 3<sup>3</sup> + 6<sup>3</sup><br /><br /><i>and for my ex-students from Japan, 259 is The number of Pokémon originally available in </i><i>Pokémon Gold and </i><i>Silver</i><br /><br />259 = 7 x 37, and there are seven primes between these two primes. <br /><br />And just in case you are dialing long distance to Zanzibar (don't we all do that on occasion) the country code is 259, and you are welcome. <br /><br />259 = 125^2 - 124 ^2, but also 22^2 - 15^2. <br /><hr /><b>The 260th Day of the Year </b><br />260 is the constant for each row, column and diagonal of the first known(1891) 8x8 bimagic square. The 8x8 is the smallest order possible for a bimagic square (the squares of the numbers also form a magic square) that uses consecutive digits. The constant for the magic square formed by the squares is 11,180. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-j1hf2erAJis/VfoJF4_MBLI/AAAAAAAAHAI/ZT01CMrqoms/s1600/bimagic%2Bsquare%2B2.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-j1hf2erAJis/VfoJF4_MBLI/AAAAAAAAHAI/ZT01CMrqoms/s400/bimagic%2Bsquare%2B2.jpg" /></a></div>More on the first of these in 1891 at the <a href="http://pballew.blogspot.com/2013/02/on-this-day-in-math-february-1.html" target="_blank">Feb 1 On This Day in Math.</a><br />260 is also the constant of Benjamin Franklin's 8x8 Magic square. <br />260 is also the sum of the squares of the divisors of 15. /( 260 = 1^2 + 3^2 + 5^2 + 15^2 /)<br /><br />260 =66^2-64^2 , and also 16^2 + 2^2 = 4^4 + 4^1, and 14^2 + 8^2<br /><br />260 is a Palindrome in base 8 (404)<br /><br />The Mayan Sacred Calendar had a 260 year cycle. <div><br /></div><div>A 5x5 magic square with a constant of 260 can be formed by taking the standard 1-25 magic square and multiplying each term by four. Another is to use 39, 40, 41,... , 64<br /><hr /><b>The 261st Day of the Year</b><br />261 = 15^2 + 6^2. It is also 45^2 - 42^2 and 131^2 - 130^2 <br /><br />261 is the number of possible unfolded tesseract patterns. (<i>Charles Howard Hinton coined the term tesseract (4-dimensional "cube"). He is also the inventor of the baseball pitching gun</i>.) (see <a href="http://pballew.blogspot.com/2011/09/baseball-and-fourth-dimension.html" target="_blank">Baseball and the Fourth Dimension</a>)<br /><br />If you draw diagonals in a 16 sided polygon, it is possible to dissect it into 7 quadrilaterals. There are 261 unique ways to make this dissection. <br /><br />261 is the only three digit number n, for which 2^n - n is prime. *Prime Curios <br /><br />261 is divisible by 9, the sum of its digits, so it is a Joy-Giver (Harshad) number. <br /><br />261 Fearless, a non-profit organization started by Kathrine Switzer, who in 1967 wore bib number 261 when she became the first woman to run the Boston Marathon as a numbered entrant. *Wikipedia<br /><br />This should be the Day of the Year in 2021, as that's the base five representation for 261(2021)<br /><br />A bracelet with 261 Blue beads and 3 Red beads can be ordered in 261 different ways. <br /><br />261 = 4^4 + 4^1 + 4^0</div><div><br /></div><div>My conjecture: There is no square that is made up of five Pythagorean triangles with a side shorter than 261, as shown *HT to @simon_gregg</div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-eQmpBiuMlKI/YHEG4Jk9VEI/AAAAAAAANdM/_BHdI2nNW5QvkFbjB4HagOBkbY-ye-iSACLcBGAsYHQ/s400/Five%2Bpythagorean%2Btriangles%2Bin%2Bone%2Bsquare.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="378" data-original-width="400" src="https://1.bp.blogspot.com/-eQmpBiuMlKI/YHEG4Jk9VEI/AAAAAAAANdM/_BHdI2nNW5QvkFbjB4HagOBkbY-ye-iSACLcBGAsYHQ/s320/Five%2Bpythagorean%2Btriangles%2Bin%2Bone%2Bsquare.png" width="320" /></a></div><br /><div><br /><hr /><b>The 262nd Day of the Year. </b> <br />262 is a palindrome, and twice a palindrome (2 x 131) <br /><br />262 is the 5th <a href="http://en.wikipedia.org/wiki/Meandric_number" target="_blank">meandric number</a>. A meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a straight road crossing a river over a number of bridges.[ The term meander is drawn from the Greek name of an actual winding river, the Maiandros.] <br /><br /> 262<sup>5</sup> begins with the digit 1234543.... novel<br /><br />262 is the number of equilateral triangles formed out of matches in a hexagonal chunk with four matchsticks on a side..(Can you find the 38 equilaterals in the hexagon with two matchsticks on a side) <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-x5Ei2nXTKMM/VfNzYsfOK2I/AAAAAAAAG-Y/vnizOZFc0Ns/s1600/matchstick%2Bhexagons.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-x5Ei2nXTKMM/VfNzYsfOK2I/AAAAAAAAG-Y/vnizOZFc0Ns/s1600/matchstick%2Bhexagons.jpg" /></a></div><br /><br />And for those "Wikipedia has everything" fans, the smallest number that does not have its own number page on Wikipedia. And yet, it is still a Happy Number. It takes only four iterations of summing the squares of the digits to get to one. <br />a Happy Number cartoon from <a href="https://twitter.com/mathhombre">John Golden, an educator well worth following, and close reading.</a> <br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-29WACeZoUdY/XyGFsQXT0NI/AAAAAAAAM7A/8LQsDHSK7TUPbG_UA1O7SnuGx0Z2akKigCLcBGAsYHQ/s1600/Happy%2BNumbers%2Bcartoon%2BJohn%2BGolden.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="900" data-original-width="629" height="640" src="https://1.bp.blogspot.com/-29WACeZoUdY/XyGFsQXT0NI/AAAAAAAAM7A/8LQsDHSK7TUPbG_UA1O7SnuGx0Z2akKigCLcBGAsYHQ/s640/Happy%2BNumbers%2Bcartoon%2BJohn%2BGolden.jpg" width="448" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*@mathhombre</td></tr></tbody></table><div class="separator" style="clear: both; text-align: left;"><br /></div><hr /><b>The 263rd Day of the Year</b><br />263 is an irregular prime. (an odd prime which divides the numerator of a Bernoulli Number) They became of great interest after 1850 when Kummer proved that Fermat's Last Theorem was true for any exponent that was a regular prime.<br /><br />\( 263^2 = 69169 \) A strobogrammatic number (appears the same rotated by 180<sup>o</sup>. *Prime Curos adds that it is the largest known number for which this is true. <br /><br />Jim Wilder@wilderlab pointed out also that the length of 263!! (Hate that notation, it is the product of all the odds from 263 down to 1) is 263 digits. Once more pushing for a different and clearer notation. Even Prime Curios \(263!_2\) is better, but doesn't allow for partial descent, so \(n!_{a,b}\) with a as step size, and b as number of steps would allow \(43!_{5, 3} = 43 * 38 * 33 \) And if you changed the lowered text to upper, the 5 could mean count UP. <br /><br />263 is the sum of five consecutive primes, 263 = 43 + 47 + 53 + 59 + 61 , and the average of the primes on each side of it, \( 263 = \frac{257 + 269}{2} \)<br /><br />263 is the sum of three primes that are all palindromes, 151 + 101 + 11, ending in the sum of the digits of 263. <br /><br />In 1919 Ramunjan wrote a new proof of Bertrand's Postulate, which he points out was first stated by Chebyshev, and I always give it in the poetic form I learned it first.<br />Chebyshev said it<br />So I'll say it again<br />There is always a Prime Between n and 2n <br />Of course Chebyshev actually said between n and 2n-2, but..... <br />He created a sequence of integers that incremented no more than one as the primes grew. That sequence of Primes, 2, 11, 17, 29, 41.. are called Ramanujan primes, and 263 is one of them. It turns out that there are 24 primes between 263 and 263/2. 131 is the 32nd Prime, 263 is the 56th. Guess Ramanujan got that one right. the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n,<br /><br />263 is also a Regular Prime. 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent p if p is regular. This focused attention on the irregular primes.[1] In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent p, <br />Srinavasa Raghava pointed out that 2+6+3 is prime, and 2<sup>x</sup> + 6<sup>x</sup> + 3<sup>x</sup> is prime when x = 3, 9, 13, 25, 27 and 29. Wonder what happens if you try that for other three digit numbers? (Don't just sit there children, get checking, and share your results with me if you find something fascinating.) <hr /><b>The 264th Day of the Year </b><br />264 = 2<sup>3</sup>x3x11 is a harshad number (a number divisible by the sum of its digits). The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The numbers were defined and named by the famous Indian Mathematician D. R. Kaprekar.<br /><br />Jim Wilder @wilderlab pointed out that the sum of all 2-digit numbers you can make from 264 totals 264... 24 + 42 + 26 + 62 + 46 + 64. <br /><br />264<sup>2</sup> = 69696, a palindrome; and 264 is the sum of ten consecutive primes. *Chaw points out that this is the only square less than 2,000,000,000 that is both a palindrome and a sum of twin primes, 34847 and 34849.<br /><br />264 is a repdigit in base 11 (222). <br /><br />264 is a Harshad (Joy-giver) number, divisible by the sum of its digits, 12. <br /><br />264 = 66^2 - 64^2 = 35^2- 31^2, <br /><br />264 is one more than a prime. <br /><br />264 is the Area of an two integer sided triangle, with sides 33, 34, 65 and 44, 37, 15.<br /><br />264 = 2^8 + 2^3, <br /><hr /><b>The 265th Day of the Year </b><br />265 is <b>!</b>6 (<i>sub-factorial 6</i>), the number of ways that six ordered objects can be mis-ordered so that each is in the wrong spot. <a href="http://pballew.net/arithme9.html#subfactor" target="_blank">See Subfactorial </a><br /><br />The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms. For example, the only derangements of {1,2,3} are {2,3,1} and {3,1,2}, so !3=2 (Pssst, students... You can find !n by dividing n! by e, and rounding to nearest integer)<br /><br />265 is the sum of two squares in two different ways, including one that is the sum of consecutive squares. \(265 = 3^2 + 16^2 = 11^2 + 12^2 \) <br /><br />265<sup>2</sup> is also the sum of two squares in two different ways, making 265 the hypotenuse of two Pythagorean Triangles. One of them is 23, 264, 265. <br /><br />265 = 16^2 + 3^2 and 29^2 - 24^2, <br /><br />265 = !6 or subfactorial 6 The number of ways of putting six letters into six addressed envelopes so that each is mis-addressed.<br /><br />265 is a semi-prime, 5 x 53. The sum of the digits of 265 is the same as the sum of the digits of its factors, sometimes called Joke numbers.<br /><br /> To form a <b>5x5 magic square</b> with a constant of 265, take the standard 5x5 using 1-25, and add forty to each term. You can also multiply all the digits from 1-25 by four, then add one, using the numbers 5, 9, 13,....., 101 </div><div><br /></div><div><br /></div><span face="system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Ubuntu, "Helvetica Neue", sans-serif" style="background-color: #f5f8fa; color: #14171a; font-size: 15px; white-space: pre-wrap;">57 64 41 48 55 63 45 47 54 56 44 46 53 60 62 50 52 59 61 43 51 58 65 42 49 </span><div>*Hat Tip to Srinivasa Ragahava<br /><hr /><b>The 266th Day of the Year</b> <br />266 can be expressed as 222 in base 11. <br /><br />266 is the sum of four cubes, \(266 = 2^3 + 2^3 + 5^3 + 5^3 \) <br /><br />It is also the index of the largest proper subgroups of the <a href="http://en.wikipedia.org/wiki/Sporadic_group" title="Sporadic group">sporadic group</a> known as the <a href="http://en.wikipedia.org/wiki/Janko_group" target="_blank" title="Janko group">Janko group</a> <i>J</i><sub>1 </sub><br /><br />266 has a digit sum of 12, a divisor of 266, so it is a Joy-Giver number. <br /><br />The sum of the divisors of 266 is 18^2 = 324. </div><div><br /></div><div>Many people know that N! has N digits for N= 22, 23, and 24. Surprisingly, to me, there are also three consecutive numbers for which N! has 2N digits, 266, 267, and 268. For N! has 3N digits, only two consecutive numbers, 2712 and 2713.For N! having 4N digits, there are again two consecutive occurrences, 27175 and 27176. For 5N we go back to three consecutive digits, <span style="background-color: white; font-family: monospace; font-size: 13px;"> </span><span style="background-color: white; font-family: monospace; font-size: 13px;">271819, 271820, 271821 </span> Note the increase by a power of ten as a limit, and the higher you go, the closer they approach e * 10^n. It has been conjectured that there are always at two or three consecutive numbers for every digit, but never more. The first 100 such numbers are found at <a href="http://oeis.org/A058814">A058814 - OEIS</a> Thanks to Derek Orr and Frank Kampas for some help and direction on this. </div><div><hr /><b>This is the 267th Day of the Year </b><br />267 = 46^2 - 43^2, and also 134^2 - 133^2<br /><br />267 is the smallest number n such that n+ a googol is prime. (anyone want to find the next one? A quick mental problem for students, How do you know that 269+Googol will not be prime?))<br /><br />267 can be written as the sum of five cubes in two ways, \( 267 = 1^3 + 2^3 + 2^3 + 5^3 + 5^3 = 2^3 + 2^3 + 2^3 + 3^3 + 6^3 \) </div><div><br /></div><div>Many people know that N! has N digits for N= 22, 23, and 24. Surprisingly, to me, there are also three consecutive numbers for which N! has 2N digits, 266, 267, and 268. <br /><hr /><b>The 268th Day of the Year.</b><br />268 is the smallest number whose product of digits is 6 times the sum of its digits. (A good classroom exploration might be to find numbers in which the product of the digits is n x the sum of the digits for various values of n.. more generally, for what percentage of numbers is the sum a factor of the product at all?)<br /><br />268 is the sum of two consecutive primes, 268 = 131 + 137 <br /><br />268 is divisible by 4, and is therefore expressible as the difference of two squares, 68^2 - 66^2<br /><br />Prime Curios offers this little mental conversion, 268 inches of 1/8 inch copper wire weighs 1 pound. There is no AWG standard gauge wire that is 1/8 of an inch diameter, but AWG 8 is really close. For students, what would a similar length of 1/4 inch diameter copper wire weigh?<br /><br />The two odd numbers adjacent to 6*268 form a pair of twin primes, and the next two odd numbers after 268 are a pair of twin primes. And the 268th prime, is the smaller of a pair of twin primes. <br /><br />And 268 is a Palindrome in base 8 (414) </div><div><br /></div><div>Many people know that N! has N digits for N= 22, 23, and 24. Surprisingly, to me, there are also three consecutive numbers for which N! has 2N digits, 266, 267, and 268. 268 is the last year date that appears in this sequence. See 266 for a link and more information. <br /><hr /><b>The 269th Day of the Year</b> <br />(on non-leap years, the 269th day is Sep 26, and the date is written 26/9 in much of Europe. This is the only day of the year which presents itself in this way. (Are there any days that work using month/day?)<br /><br />269 is a regular prime, an Eisenstein prime with no imaginary part, a long prime, a Chen prime, a Pillai prime, a Pythagorean prime, a twin prime, a sexy prime, a Higgs prime, a strong prime, and a highly cototient number. So many new terms to look up... Well? Look them up.<br /><br />269 is the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix <br /><br />Prime Curios offers this interesting convention of prime numbers, "The longest official game of chess on record (269 moves) took place in Yugoslavia on 2/17/89 and ended in a draw. Note that 2, 17, 89, and 269 are all prime numbers." And I'm guessing that Yugoslavia was a Prime country in its day. <br /><br />Prime Curios also had this interesting tidbit, "The smallest prime whose the square, 72361, is a concatenation of primes in two ways, i.e., (7, 23, 61) and (7, 2, 3, 61).)<br /><br />269 is the largest prime factor of 9! + 1 = 362881. <br /><br />269 is a Pythagorean Primes (of the form 4n+1) are the sum of two squares, conjectured by Fermat, proved by Euler. 269 = 10^2 + 13^2. <br /><br />269 is also the Hypotenuse of a Primitive Pythagorean triple, (69, 260 269) <br /><br />269 is also the difference of two squares, as 235^2 - 234^2, <br /><br /><hr /><b>The 270th Day of the Year</b> <br />the harmonic mean of the factors of 270 is an integer. The first three numbers with this property are 1, 6, and 28 (which are all perfect #s).. what is the next one? Often called harmonic numbers, they are sometimes called Ore numbers for Øystein Ore, who studied them, and showed that all perfect #s are harmonic. . Many of them also have the arithmetic mean of their divisors is an integer, but not all.<br /><br />270 is the sum of eight consecutive primes, 270 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 ; and the sum of three cubes \(270 = 3^3+ 3^3 + 6^3 \). <br /><br />10! = 3628800 has 270 factors. (A really good high school student should be able to confirm this quickly.)<br /><br />270, with a digit root of nine, and a digit sum of nine, is divisible by the sum of its digits and thus, a Joy-Giver (Harshad) number. <br /><br />270 is also the smallest positive integer with divisors ending in each of the digits 1- 9, divisible by 1, 2, 3, 54, 5, 6, 27, 18 ,9,.<br /><br />and of course, between a pair of twin primes, it is the average of two primes.<br /><br />270 has only three prime factors, the consecutive primes 2, 3, and 5.</div><div><br /></div><div>270 is the fifth, and largest year day which has an integral Harmonic mean for the divisors (including n) of the number. These are usually called Harmonic Divisor numbers to distinguish them from the Harmonic numbers (they are sometimes called Ore numbers after Oystein Orr). All perfect numbers have this property, 6 and 28 for example, and my experience is that all perfect numbers have a HM which is prime but the converse is not true. The harmonic mean of the divisors of 270 is six, but the harmonic mean of 140 is 5, the same as the harmonic mean of the divisors of 496, the next perfect number after 28. The first five perfect numbers have harmonic means of their divisors of 2, 3, 5, 7, and 13. The harmonic mean of the divisors of a perfect number is the index of the Mersenne prime which is a factor of the number , 496 = \(2^4 x (2^<b>5 </b>- 1) \) and the harmonic mean of its divisors is 5.<br /><hr /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-80357057323059857672020-07-16T08:20:00.022-07:002021-03-07T09:56:58.697-08:00Number Facts for Every Year Date, 211-240<b> The 211th Day of the Year</b><br />The 211th day of the year; 211 is a primorial prime,(a prime that is one more, or one less than a <a href="http://en.wikipedia.org/wiki/Primorial" target="_blank">primorial</a> can you find the next larger (or smaller) primorial prime? <br /><br />211 is also the sum of three consecutive primes (67 + 71 + 73)...<br /><br />There are 211 primes on a 24-hour digital clock. (00:00 - 23:59) *Derek Orr @ Derektionary 211 is the 4th** Euclid number: 1 + product of the first n primes.(after Euclid's method of proving the primes are infinite. most Euclid numbers, unlike 211, are not themselves prime, but are divisible by a prime different than any of the primes in the product n#) (**some would call it the fifth since Euclid seemed to consider 1 as a unit as similar to the primes.)<br /><br />211 is a prime lucky number, and there are 211 lucky primes less than 10^4 (or 10 ^(2+1+1))*Prime Curios<br /><br />211 is the concatenation of the smallest one digit prime and the smallest two digit prime, 2, 11.<br /><br />211 = 3^5 - 2^5, two consecutive fifth powers, it is only the second, following 31, and is the last year date with the property.<br /><br />Hardy wrote a New Year Resolution in a card to Ramujan to get 211, none out, in a cricket test match at the oval. <br /><br />A Lazy Caterer number, A Pizza can be cut into 211 pieces with 20 straight cuts. <br /><br />211 is a repunit in base 14 (111)14^2 + 14 + 1 <br /><br />211 is also SMTP status code for system status.*Wik <br /><br />211 is an odd number, so it is the difference of two consecutive squares, 106^2 - 105^2 = 211 211 is the minimum sum of any row, column, or diagonal, of a minimum difference (not all rows and columns are the same, but the difference between largest, 213, and smallest, 211, sum is smaller than any other option) prime magic square that contains the 25 primes less than 100 ^Prime Curios<br /><br />41 79 17 13 61<br />53 03 83 67 07<br />59 97 05 23 29<br />11 31 37 89 43<br />47 02 71 19 73<br /><br />211 is the first of fifteen consecutive odd numbers that sum to the cube of 15, 3375<br /><br />211 is a prime of the form 4k+3. According to Guass' reciprocity law, if two numbers, p and q are in this sequence then there exists a solution to only one of x^2 = p (mod q) or x^2 = q (mod p). 3 is another number in the sequence. Can you find an x^2 so that one of these congruences is true?<br /><br />And one more from *Prime Curios. If you've ever heard the expression "a month of Sundays," for something that takes a really long time that's 31 Sundays, starting on a Sunday and going for 30 more weeks to end on a Sunday, or 211 days, Sunday to Sunday. <br /><hr /><b>The 212th Day of the Year</b>Besides being the Fahrenheit boiling point of water at sea level, 212 produces a prime of the form k<sup>10</sup>+k<sup>9</sup>+...+k<sup>2</sup>+k+1, when k=212. Edward Shore@edward_shore sent me a note:" That number would be 184,251,916,841,751,188,170,917.") (<i>students might explore different values of k, and different maximum exponents to produce primes..ie when k is 2, then 2<sup>6</sup> +2<sup>5</sup>+...+2<sup>2</sup>+2+1 is prime</i>) <br /><br />The smallest even three-digit integer, abc, such that (abc)/(a*b*c) is also prime. [ie 212/(2*1*2)= 53 ]*Prime Curios<br /><br />212 is a palindrome whose square is also a palindrome, 212<sup>2</sup>= 44944. It is the last year date for which this is true. <br /><br />212 a palindrome is the average of two emirps, 113 and 311 <br /><br />212 is a palindrome in base ten, and also in ternery (base 3, where it is 21212), <br /><br />212 is a balanced binary number, with the same number of ones and zeros. 11010100. That means it is one path from (0,0) to (4,4) ,moving only up or right on the lattice. There are 70 such possible paths, How many are always on or above the line y=x? <br /><br />212 in base three is 14 (why does that sound so familiar?), but in base 7 and base 9 it is a prime (212_7) = 107; (212_9) = 173 <br /><hr /><b>The 213th Day of the Year</b><br />213 is a square free number as it has no repeated prime factors. How many days of the year are square free?<br /><br />For 213, the sum of the digits and the product of the digits are equal, and forms a prime when one is added or subtracted.<br /><br />The average of the prime factors of 213 is a prime number. [213 = 3*71 and (3+71)/2=37 ]<br /><br />The square of 213 is a sum of distinct factorials: 213<sup>2</sup> = 45369 = 1! + 2! + 3! + 7! + 8!, it is the smallest 3 digit number with this property. (what's next?)<br /><br />213, like all odd numbers, is the difference of two consecutive squares, 107^2 - 106^2, and because 6*34+9 = 213, 213= 37^2 - 34^2,<br /><br />a 3x3 magic square with magic constant of 213 using consecutive integers<br />74 67 72<br />69 71 73<br />70 75 68<br /><br />And with a Hat Tip to John Golden@MathHombre, who designed the google sheet, here is a combination of Latin Squares to form a 3x3 magic square with a constant of 213 (see day 177 for link to his pages)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-eAXzljVG6iw/XvUbJz0Rh-I/AAAAAAAAMx0/bXwKos_93DM5nmL2cComxYVKjIFOrXG5ACLcBGAsYHQ/s1600/Latin%2BMagic%2Bsquare%2B213.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="254" data-original-width="317" height="256" src="https://1.bp.blogspot.com/-eAXzljVG6iw/XvUbJz0Rh-I/AAAAAAAAMx0/bXwKos_93DM5nmL2cComxYVKjIFOrXG5ACLcBGAsYHQ/s320/Latin%2BMagic%2Bsquare%2B213.jpg" width="320" /></a></div><br /><br />213^2 = 45369 = 1! + 2! + 3! + 7! + 8!. There are only 11 days that have this property. 213 is the second largest. *HansreudiWidmer. <br /><br /><hr /><b>The 214th Day of the Year</b><br />The 11th perfect number 2<sup>106</sup> (2<sup>107</sup>−1) has 214 divisors. Also, 214*412+1 is prime. *Prime Curios<br /><br />214 is the middle number in a string of three consecutive semiprimes <br /><br />214 is the last day of the year for which n!!-1 is prime, it is a 205 digit number ending in 25 consecutive nines. (The N!! symbol is often confused with (n!)! in which (3!)! = 6! that it should not be used by teachers. This usage means 214*212*210..*2, and if you wanted every fifth term, you could write n!!!!! which becomes useless at 13, or 43. I much prefer an adjustment of Kramp/ Vandermonde method which I write as \( 214!_2\) but if you only wanted, say the 214 * 212 * 210 * 208, you could add \( 214!_{4|2}\) <br /><br />You can (sort of) cancel out common factors in a Choose function. <br />\( \binom{321,214} =\binom{107 * 3, 107 * 2}=\binom(3,2} = 3 \). Ok that's way wrong, the actual answer (below) is 88 digits long; but it is right in mod 107 (the cancelled term). *HT to John Cook<br />binomial(321, 214) Result: 2566686889891552749675710089869130253760467178930978799676532656352437721086512214025600 <br />Ok, that's a really short list. HELP, what am I missing. Send me your favorite 214 facts. <br /><hr /><b>The 215th Day of the Year</b><br />There are 215 sequences of four (not necessarily distinct) integers, counting permutations of order as distinct, such that the sum of their reciprocals is 1. Obviously, one of them is 1/4+1/4+1/4+1/4=1. How many can you find?<br />If you get stuck, look at the bottom of this Math Day<br />How many solutions with four distinct integers, not counting permutations? <br /><br />215 in base six is a repdigit, 215<sub>[10]</sub> = 555<sub>[6]</sub><br /><br />Lagrange's theorem tells us that each positive integer can be written as a sum of four squares, but Lagrange allowed the use of zeros, such as 1<sup>2</sup> + 1<sup>2</sup> + 1<sup>2</sup> + 0<sup>2</sup> =3. Allowing only positive integers, there are 57 year days that are not expressible in less than four squares. 215 is the 34th of these year days that is NOT expressible with less than four positive squares. 215 = 1<sup>2</sup> + 3<sup>2</sup> + 6<sup>2</sup> + 13<sup>2</sup>. <br />The last such number was 207, the next is 220.<br />You can create more on your own. Take any number in the sequence and multiply it by an odd number and you have another that is in the sequence. <br />In 1937 Lothar Collatz conjectured the process of starting with any number and repeatedly multiplying by 3n+1 if odd, or dividing by two if even in iteration will always lead to one. I mention that here because if you start with 215, it will take 101 operations (a prime number) before you get back to one.<br /><br />215 = (3!)^3-1 *Wik<br /><br />215 is the second (and last) yearday that N^2 - 17 is a square. The next such number is over 4000, but can you find the smaller?<br /><br />Like all odd numbers, 215 is the difference of two consecutive squares, 108^2 - 107^2 = 215. Because it ends in five (and is bigger than 35) it is also the differences of two squares of numbers that differ by 5, 24^2 - 19^2 = 215<br /><br />All twin primes after 3 are of the form 6n-1 and 6n+1. A pair of twin primes are formed by 6(215)+1 and 6(215)-1 <br />A nice foot note to this fact is that the 215th and 216 primes are twin primes.<br /><br />215 is the sum of discrete factorials, 8! + 7! + 6! + 5! + 4! + 1!. <br /><br />215 is the difference of two cubes, 6^3 - 1^3. <strike></strike><br />24 arrangements of (2,3,7,42), (2,3,8,24), (2,3,9,18), (2,3,10,15), (2,4,5,20) and (2,4,6,12).<br />12 arrangements of (3,3,4,12), (3,4,4,6), (2,3,12,12), (2,4,8,8) and (2,5,5,10).<br />6 arrangements of (3,3,6,6).<br />4 arrangements of (2,6,6,6).<br />1 arrangement of (4,4,4,4).<br /><hr /><b>The 216th Day of the Year</b><br />216= 6^3 = 2^3 + 3^3; I'm calling such numbers cube-full, in analogy to square-full numbers, (see 196) since every prime p that divides 216,p^2 and p^3 will also divide 216. Only one more year day is a perfect cube.<br /><br />According to<b> *Derek Orr</b>, m^k is the largest number n such that (n^k-m)/(n-m) is an integer (for k > 1 and m > 1). So no number n larger than 216 will make (n^3 - 6 )/(n-6) is an integer. When n = 216, the quotient is 47989 (Hope I got that right).<br /><br />Because 216/4=54, it must be the difference of two squares, 55^2-53^2 , and because 216/8 = 27, 216 = 29^2 - 25^2 <br /><br />Also, because 8*25 + 16 = 216, it is the difference of two squares of the numbers form n and n+4; in this case 29^2 - 25^2=216. (two different rules that gave the same solution. and because 216 = 12 * 18, 216 = 21^2 - 15^2<br /><br />Since 216 = 3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup> = 6<sup>3</sup>, it is the smallest cube that's also the sum of three cubes (Plato was among the first to notice this, and mentioned it in Book VIII of Republic). (What is the next cube that is the sum of three cubes?... and how can you be sure there will never be a day that is a cube that is the sum of two cubes?) <br /><br />According to the Ken Burns series Baseball, 216 is the number of stitches on a baseball.*Wikipedia<br /><br />Zhi-Wei Sun conjectured in March 2008 that 216 is the only number not of the form p + k(k+1)/2, with p = 0 or p prime. *Prime Curios <br /><br />6 of one and a dozen and a half of another, The numbers 6*216+1, 12*216+1, and 18*216+1 are all prime *Prime Curios<br /><br />216 is a balanced binary number, having four ones, and four zeros. <br /><br />216 is a palindrome in base five, (1331) =5^3 + 3*5^2 + 3 * 5 + 1 =216<br /><br />216 is also the sum of a twin prime pair (107 + 109).<br /><br />A multiplicative Magic Square with a constant of 216.<br />2 9 12<br />36 6 1<br />3 4 18<br />*Wikipedia<br /><br />The smallest and only known cube sandwiched between two triplets of semiprimes (i.e., 213=3*71, 214=2*107, 215=5*43 and 217=7*31, 218=2*109, 219=3*73). *Prime Curios<br /><br />There are 216 fixed hexominoes, the polyominoes made from 6 squares. *Wikipedia<br /><br />And for those who are interested in variations on 666, the so called number of the beast, \( 6^{(6^6)} =216 \) mod 360 from which we get the divine equivalent, \( cos(216)^o= \frac{- \phi}{2} \)<br /><br />Ok, one more 216 to 666 connection from Ben Vitalle. There is a right triangle with legs of 216, 630 and a hypotenuse of (wait for it...... ) 666 Benjamin Vitale @BenVitale <br />216 is the smallest number that may be expressed as the difference of the sums of the squares of successive twin primes (13^2 + 11^2) - (7^2 + 5^2).*Prime Curos <br /><hr /><b>The 217th Day of the Year</b><br />217 is both the sum of two positive cubes and the difference of two positive consecutive cubes in exactly one way: 217 = 6<sup>3</sup> + 1<sup>3</sup> = 9<sup>3</sup> − 8<sup>3</sup>. (How frequently would the difference of two consecutive cubes also be expressible as the sum of two cubes?)<br /><br />217 is the sum of four cubes, 1^3 + 3^3 + 4^3 + 5^3 <br /><br />217 is a palindrome in base six, (1001) and in base 12, (161).<br /><br />Anti-sigma(n) is not a well known function, but anti-sigma(22) = 217. Simply add all the numbers from one to 22, then subtract all the numbers that divide into 22 evenly: [ (22*23)/2 - 1 - 2 - 11 - 22 = 217 ]<br /><br />217 = 109^2-108^2 = 19^2-12^2<br />217^2 = 47089. Both 47 and 89 are primes, as is their concatenation, 4789. *Prime Curios<br /><br /><br />217 = 7 x 31, the first of three consecutive semi-primes. <br />A 7x7 Magic square with integers 7 - 53 has a magic sum of 217 *@SrinivasR1729 <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-i6OmZElicFg/XynQw8m0QBI/AAAAAAAAM7o/NOK4VBAXfJsIWiHGyfCjlvOOoOOBQfm2ACLcBGAsYHQ/s1600/magic%2Bsquare%2B7x7%2B217.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="334" data-original-width="461" height="290" src="https://3.bp.blogspot.com/-i6OmZElicFg/XynQw8m0QBI/AAAAAAAAM7o/NOK4VBAXfJsIWiHGyfCjlvOOoOOBQfm2ACLcBGAsYHQ/s400/magic%2Bsquare%2B7x7%2B217.jpg" width="400" /></a></div><br />217 is the ninth centered hexagonal number.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-3AJZRO1nhro/XwUN1lGitbI/AAAAAAAAM5M/7YYCaz0xuI84lhUtJ7jHw7Sut-2hRtM8ACLcBGAsYHQ/s1600/centered%2Bhexagonal%2Bnumbers.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="145" data-original-width="307" height="188" src="https://1.bp.blogspot.com/-3AJZRO1nhro/XwUN1lGitbI/AAAAAAAAM5M/7YYCaz0xuI84lhUtJ7jHw7Sut-2hRtM8ACLcBGAsYHQ/s400/centered%2Bhexagonal%2Bnumbers.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">OEIS</td></tr></tbody></table><br /><hr /><b>The 218th Day of the Year </b><br />109 is the sum of two squares, 10^2 + 3^2. Can you see how to use this to get the sum of squares of 2x 109?<br />218 = 7<sup>2</sup> + 13<sup>2</sup> <br /><br />218 = 6^3 + 1^3 + 1^3, and the difference of two cubes, 7^3 - 5^3.<br /><br />218 is the number of nonequivalent ways to color the 12 edges of a cube using at most 2 colors, where two colorings are equivalent if they differ only by a rotation of the cube.<br /><br />218 is the smallest number with a Merten funtion =3. (an acceptable definition for students is that the Merten number for n, M(n), is the count of square-free integers up to n that have an even number of prime factors, minus the count of those that have an odd number.) The function is named in honor of Franz Merten, who was a teacher of Schrodinger.<br /><br />218 is the number of points on a 6x6x6 space lattice<br /><hr /><b>The 219th Day of the Year</b><br />219 is an odd number, so it is the difference of two consecutive squares, 219 = 110^2 - 109^2, and because 219 = 6n+9 for n=35, then 219 = 38^2 - 25^2<br /><br /><br />The Merten Function of 219 (see 218) = 4, a second day hitting a record high. <br /><br />There are 219 space groups in 3 dimensions, analogous to the 17 wallpaper groups in 2 dimensions. <br /><br />219 <sup>p</sup> +2 is prime when p is any of the first three primes. Srinivase Raghava adds that it is also true with exponets of 6, 23, 34, 35, 36, and 64. <br /><br />219 is a palindrome in binary, (11011011), and a repdigit in base 8 (333). And I think it is kind of cute that in base 36, it is (63)<br /><br />219 is the sum of four cubes (not all distinct) and in more than one way. One way is 6^3 + 1^3 + 1^3 + 1^3, can you find the other?<br /><br />There are 219 ways to partition 37 into prime parts. <br /><br />219 is a Happy Number, the iteration of the sum of the squares of the digits eventually maps to one. <br /><br />219 is the smallest number that is expressible as the sum of four possible cubes, in two different ways. <br />the other sum of four cubes is 3^3 + 4^3 + 4^3 + 4^3. ,br. <br /><hr /><b>The 220th Day of the Year</b><br />220 is the smallest amicable number, paired with 284. Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other. Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. (<i>what is the next pair?</i>)<br /><br />If you add the sum of all the divisors of the first 16 numbers, you get 220.<br /><br />220 is the the largest difference between two consecutive primes less than 100,000,000.<br /><br />220 is the tenth tetrahedral number, the sum of the first ten triangular numbers, 1 + 3 + 6 + ... + 55 *Wikipedia <br /><br /><br />Finding a number whose reciprocal is equal to the sume of two other reciprocals is an important idea in many areas of math and science. 220 turns out to be the largest of a triple of that type, 1/220 + 1/180 = 1/99 <br />Because 220/4 = 55, 220 = 56^2 - 54^2, If all the diagonals of a regular dodecagon are drawn, they divide the dodecagon into 220 regions. <br /><br />220 is the sum of four consecutive primes. 47 + 53 + 59 + 61. <br /><br />Interesting that if you multiply the first 220 composite numbers, it is a composite number that falls between a pair of twin primes. *Prime Curios <br /><br />Every number less than 220 can be formed by the sum of divisors of 220 . <br /><br /> 220 is the largest gap between consecutive primes less than 10^8.<br /><br /> And <b>Derek Orr</b> Points out that the aliquot(proper divisors) sequence for 220 does not end in 1. This was known to the ancients who described such numbers as amicable numbers. 220 and 228 are each the aliquote sum of the other. There are a few other numbers which have repeating sequences of three or more in a loop. The aliquot sequence of perfect numbers is fixed by their definition. There are other non-perfect numbers whose sequence eventually lands on a perfect number and then repeats that number infinitly (are until you stop calculating). and there are a few other numbers which have repeating sequences of three or more in a loop. <br /><br />You can make a 5x5 magic square with consecutive numbers from 32 to 44, following the order of the standard 5x5 magic square.... or simply add 31 to each one in the primitive square. <hr /><b>The 221st Day of the Year </b><br />221 represents the hypotenuse of four Pythagoren triangles, (21^2+220^2=221^2), (85^2+204^2=221^2), (104^2+195^2=221^2), (140^2+171^2=221^2). 221 is also expressible as the sum of two squares in two different ways: (5^2+14^2 = 221 = 10^2+11^2). *Prime Curios <br /><br />If you deal two cards at random from a standard deck, your chances of getting two aces, is 1 in 221. <br /><br />221 the sum of consecutive prime numbers in two different ways 221 = (37 + 41 + 43 + 47 + 53) = (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41)<br /><br />And of course, 221 is the product of consecutive primes, 13 x 17 <br />Because 13 and 17 are both 4n+1 primes, they are the sum of two squares (13= 3^2 + 2^2^2 and 17 = 4^2 + 1^2), these can be used to construct two ways that 221 can be shown as the sum of two primes. (221 = (3*4+1x2)^2 + (4x2-1x3)^2 = 14^2 + 5^2 and by changing +/- (3x4-1x2)^2 + (4x2+1+3)^2 = 10^2 X 11^2 = 221. Known around 1st century AD by Diophontas. <br />221<sup>221</sup> + 122 is prime, it is the <i>only</i> known number greater than one with this property. <br /><br />221 = 111^2 - 110^2.<br /><br />221 is the number of 7-vertex Hamiltonian planar graphs ( a graph that allows a closed path that visits each node exactly once.)<br /><br />2*3*5*7+11=13*17. Note the consecutive use of the first 7 primes. *Prime Curios <br /><br />221 in base 16 is given by (DD) or (13, 13) 13*16 + 13. <br /><hr /><b>The 222nd Day of the Year </b><br />A repdigit, 222 = 2 x 3 x 37<br /><br />222 is called a sphenic (Greek for wedge) number. They have three distinct prime factors. 30= 2x3x5 is the smallest sphenic number. Can you find two consecutive numbers that are both sphenic numbers? More??<br /><br />481 / 222 = (4+8+1)/(2+2+2) = 13/6 *Potetoichiro<br /><br />222 = (3!)<sup>3</sup> + (2!)<sup>2</sup> + (1!)<sup>1</sup> + (0!)<sup>0</sup> *Derek Orr @Derektionary<br /><br />222 is the number of lattices on 10 unlabeled nodes. <br /><br />and 222 is the sum of consecutive primes. 109 + 113, No larger year date shares this quality<br /><br />222 is the smallest repdigit such that the product of itself and all truncations of itself plus and minus one results in twin primes. I.e., 222*22*2 ± 1 are twin primes. *Prime Curios <br /><br />The digit sum of 222 in base ten is the same as digit sum in binary, and also in base three. <br /><br />222 is the sum of all two digit primes formed by consecutive digits. *Prime Curios <br /><br />222 is also a repdigit in base 36 (66) <br /><br />222 is called a pseudo-perfect number, it is the sum of a subset of its divisors, 37+74 + 111 = 222 <br /><hr /><b>The 223rd day of the Year</b><br />223 is the 48th prime number, formed from three consective prime digits, and the sum of three consecutive primes (71 + 73 + 79), 223 and also the sum of seven consecutive primes (19 + 23 + 29 + 31 + 37 + 41 + 43)<br /><br />Every number can be formed with no more than 36 fifth powers, except one, 223 is the only number that requires 37 fifth powers. This is related to Waring'a problem. In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers to the power of k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. <br /><br />Fans of Star Wars may know that this is sometimes called the Star Wars Droid Prime because it uses only the numbers in the names of R2-D2 and C3PO <br /><br />An interesting note, if you take a prime number less than 223 and reverse it, some are prime, some have two factors, but 223 is the smallest prime whose reversal has three factors (322 = 2 * 7 * 23) <br /><br />223 is the difference of consecutive squares, 112^2 - 111^2<br /><br />The number of primes and the number of composites that cannot be written as the sum of two primes, up to 223, are equal.*Prime Curios <br /><br />The prime preceding 223 The sums of the nth powers of its digits are prime for all n between 1 and 6 inclusive: sum of digits = 7, sum of squares of digits = 17, sum of cubes of digits = 43, sum of fourth powers = 113, sum of fifth powers = 307 and sum of sixth powers = 857. *Prime Curios <br /><br />If you take the tens compliment of the digits of 223, you get 887, another prime. The same is true for the next three primes following 223. <br /><br />If you take the square of 223 which is 49729 observe that the last three digits, 729 are prime. Try any of the next dozen primes following 223 and you will observe the same result. *Prime Curios <br /><br />223 sets a new high for the distance between the two nearest primes surrounding it, they are 16 units apart. *Wikipedia <br /><br /><hr /><b>The 224th Day of the Year</b><br />224 is the sum of the cubes of 4 consecutive integers:<br />224 = 2<sup>3</sup> + 3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup> and also 2<sup>3</sup> + 6<sup>3</sup><br /><br />Cool thing about 224: 224 = 23+45+67+89 *Derek Orr @Derektionary <br /><br />Every number smaller than 224 can be expressed as the sum of distinct divisors of 224. <br /><br />224 = 2^5 x 7. With so many factors of two, it has lots of representations as the difference of two squares. 224 = 57^2 - 55^2 = 30^2 - 26^2 = 15^2 - 1^2<br /><br />224 is a palindrome in base 3 (22022).<br /><br />14 red checkers and 14 black checkers on a string of 15 squares can be exchanged by sliding, or jumping in 224 moves. <br /><br />224 is one more than a prime, and one less than a square, and its square, 50176, is one less than a prime. <br /><hr /><b>The 225th Day of the Year</b><br />225 = 01 + 23 + 45 + 67 + 89 *HT to Derek Orr <div><br /></div><div>Playing around with numbers n, such that n^k has a digit sum of n, turns out that 225^21 = 24878997722115027320114677422679960727691650390625, and the digit sum, 225. <br /><br />225 is the ONLY three digit square with all prime digits. Can you find a four digit square with all prime digits?<br /><br />225 = (3!)<sup>3</sup>+(2!)<sup>3</sup>+ (1!)<sup>3</sup><br /><br />\(225 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 \) (which means, of course, that \( 225 = (1+2+3+4+5)^2 \) *Derek Orr @Derektionary It is the last year day that is the sum of five consecutive cubes. <br /><br />225^n + 2 is prime for half the values of n from 1 to 10. *Prime Curios <br /><br />225 is the smallest number that is a polygonal number in five different ways.[1] It is a square number (225 = 152),[2] an octagonal number, and a squared triangular number (225 =15^2 = (1 + 2 + 3 + 4 + 5)^2 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3) *Wikipedia (Student note. The square of the nth triangular number is always the sum of the first n cubes. For example 1^2 + 2^2 + 3^3 = (1+2+4=3)^2 = 36. <br /><br />225 is the smallest square that can be can be written as the sum of a cube and a square (a^3 + b^2) in two ways, i.e., 225=5^3+10^2=6^3+3^2*Prime Curios<br /><br />I'm still pushing for the symbol n!<sub>2<\sub>; for (n * (n-2) * ... what some call the double factorial. So \(225 = (5!_2)^2 =(5 * 3 * 1)^2\)<br /> Also (4^2 - 1) ^2 </sub><br /><br />If you take the normal 3x3 magic square, and multiply each digit by 15, you get a magic square with a constant of 225. Or just add 70 to each value and get a different one using 71-79 with 225 for the magic constant <br /><br />\(225 = 15^2 , and 225 = 25^2 - 20^2 = 113^2-112^2 = 39^2 - 36^2 \)<br /><hr /><b>The 226th Day of the Year</b><br />The iteration of the sum of the squares of the digits leads to one (a happy number). What percentage of numbers have this property?<br /><br />226 = 3!<sup>3</sup>+2!<sup>3</sup>+1!<sup>3</sup> + 0!<sup>3</sup> *Derek Orr <br /><br />The binary expression for 226 has the same number of ones and zeros. There are only 49 such year days, and this is number 46.<br /><br />226 is one more than a square, and one less than a prime.<br /><br />226 =15^2 + 1^2. So what might you learn from seeing that written as (8+7)^2 + (8-7)^2?<br /><br />226 has a totally balanced binary expression, with four ones and four zeros.<br /><br />226 is the tenth centered pentagonal number. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-OC1jODpF9rU/XwizzGA4wzI/AAAAAAAAM5k/15B0Vfn5N_wsC03UTIbavc2dWlPt1jQ8QCLcBGAsYHQ/s1600/centered%2Bpentagonal%2Bnuber.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="98" data-original-width="240" height="131" src="https://2.bp.blogspot.com/-OC1jODpF9rU/XwizzGA4wzI/AAAAAAAAM5k/15B0Vfn5N_wsC03UTIbavc2dWlPt1jQ8QCLcBGAsYHQ/s320/centered%2Bpentagonal%2Bnuber.png" width="320" /></a></div><hr /><b>The 227th Day of the Year</b><br />227 is a prime number, and the smaller of a twin prime pair; but it can also be written as the sum of the sum and the product of the first four primes: (2 + 3 + 5 + 7)+(2 x 3 x 5 x 7) = 227.<br />In a similar way, the first two primes work (2+3)+(2x3)=11 is prime. Can you find another? (Ben Vitale has found all the cases under 1000 for which p = (a + b + c + … ) + (a * b * c …) He even found another way to express 227. <a href="http://benvitalenum3ers.wordpress.com/2012/08/15/prime-num3ers-that-can-be-expressed-as-ab-ab/" target="_blank">His blog</a> also has lots of other number curiosities, so give it a look. Much fun.<br /><br />227 is also the largest odd day number of the year which can NOT be expressed as a prime added to twice a square. There are three others you might fine, and three others larger than 366. Observe that these are all seven prime.[OEIS gives ten numbers that include 1, 5779, and 5993. These last two are composite. <i>and that seems to be ALL of them that exist. if there are more, they are larger than 10^13.</i>)<br />This problem is based on an original conjecture by C Goldbach that all ODD COMPOSITE numbers could be written as twice a square plus a prime. The laxt two show he was wrong.<br /><br />227 is the 7^2 prime number *Prime Curios<br /><br />The harmonic sequence, or sum of the reciprocals of the integers grows to infinity, but slowly. It takes the 227th term (1/227) to finally push it over the value 6.(And don't even think about trying to get to seven!)<br /><br />A beauty about six primes, 227 + 251 + 257 = 233 + 239 + 263, and if you square each one, 227^2 + 251^2 + 257^2 = 233^2 + 239^2 + 263^2 *Prime Curios. <br /><br />227 is a palindrome in base eight (343)<br /><br />The number (7 * 10^227+71)/3 Forms a prime with the digit 2 followed by 225 digits of 3, then ending in 57 <br /> Students might try a few starting with 3, ,4, etc.<br /><br />22/7 is a common approximation for Pi in middle school.<br /><br />There are 227 composite days in a year. *Prime Curios <br /><hr /><b>The 228th Day of the Year</b><br />228 is the number of ways, up to rotation and reflection, of dissecting a regular 11-gon into 9 triangles.<br /><br />228 + 1, 822 + 1, and (228 + 822) + 1 are all primes. *Prime Curios <i>Is there another such year day?</i><br /><br />228 in binary is written 11100100 notice that this is all four possible two digit binary combinations in descending order, 11, 10, 01, 00. (Just figured out that the equivalent in base three is a little over 10<sup>39</sup>)<br /><br />228/4 = 57 so 228 = 58^2 - 56^2. <br /><br /> 228 is a palindrome in base 12 (171). (I resist using "duodecimal system" because I once mis-spoke in class and said "Dewey decimal system" twice in one brief comment. My students that year seemed to have thousands of comments about trivia related to the actual Dewey Decimal system that they would share with each other in class.) <hr /><b>The 229th Day of the Year</b><br />229 is the 50th prime, and is the smallest prime that added up to the reversal of its digits yields another prime, (229 + 922) = 1151 (can you find the next one?)<br /><br />The sum of the digits of 229 is prime (13) and the sum of squares of the digits is also prime (89).<br /><br /> extra: 229 is the difference between 3³ and 4⁴ *jim wilder @wilderlab <br /><br />If you replace each digit of 229 with its square, you get a prime number, 4481. If you do the same with its cube, you get 88729, another prime. *Prime Curios<br /><br />1/229 has 228 digits in its period<br /><br />If you replace each digit with its ten complement, you get 881, another prime.<br /><br />100! + 229 is prime<br /><br />2^229 is a 69-digit number containing only one zero. Is this the largest power of two that has one or more unique digits? *Prime Curios <br /><br />If you draw K13, the complete graph with 13 vertices so that it has the fewest possible crossings, it will still have 229 crossings *Wikipedia<br /><br />1/229 has 228 digits in its period<br /><br /> <hr /><b>The 230th Day of the Year</b> <br />230 is the smallest number such that it and the next number are both sphenic numbers, the product of three distinct primes (230 = 2*5*23 and 231 = 3*7*11). *Prime Curios (can there be three consecutive numbers that are the product of three (or n) distinct primes?<br /><br />Their are 230 possible crystal shapes that can tile space (this counts the chiral reflections as separate). This is the analogy of the better known 17 "wallpaper groups" which tile the plane. Interestingly, both were proved by the same man, Evgraf Fedorov, in 1891. He did the plane problem the hard way, he proved the case for space first, then worked backwards to the plane. <br /><br />230 is a Harshad number, divisible by the sum of its digits <br /><br />And 230 is another Happy number.<br /><br />The sum of the proper divisors of 230 is 202. Iterating this process on each new number gives 104, 106, 56, 64, 63, 41, 1, 0, <br /><br />230^2 + 1 = 52901, a prime<br /><hr /><b>The 231st Day of the Year</b><br />there are 231 cubic inches in a US Gallon, (admit it, you did NOT know that.) <br /><br />Ok, and it's also the sum of the squares of four distinct primes, 231 = 2<sup>2</sup> + 3<sup>2</sup> + 7<sup>2</sup> + 13<sup>2</sup>.<br /><br />\((3!)^3 + (2!)^4 - (1!)^5 \)<br /><br />231 = 12 + 23 + 34 + 45 + 56 + 61 (loop 1-2-3-4-5-6-1)<br />231 = 98 + 76 + 54 + 3<br />*Derek Orr<br /><br />231 = 40^2-37^2 = 20^2 - 13^2<br /><br />231 in base twenty seems as small as a BB. (11 x 20 + 11)<br /><br />The average number of distinct prime divisors for all n less than a googolplex is only about 231. *Prime Curios <br /><br />The sum of the first 21 integers, 231 is the 21st triangular number, It is also a hexagonal and an octahedral number.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-UGhNBrM0H38/XwjfqrRcAKI/AAAAAAAAM5w/MSCY0QEQ18UESDGeLwz6d63Vvy4y4oXKACLcBGAsYHQ/s1600/Octahedral_number.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="220" data-original-width="220" height="320" src="https://2.bp.blogspot.com/-UGhNBrM0H38/XwjfqrRcAKI/AAAAAAAAM5w/MSCY0QEQ18UESDGeLwz6d63Vvy4y4oXKACLcBGAsYHQ/s320/Octahedral_number.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Wikipedia</td></tr></tbody></table><br /><br />Their are 231 integer partitions of 16.<div><br /></div><div>Here are the seven partitions of 231 into strings of consecutive counting numbers </div><div><br /></div><div><span face="system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Ubuntu, "Helvetica Neue", sans-serif" style="background-color: #f5f8fa; color: #14171a; font-size: 15px; white-space: pre-wrap;">231=1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21 231=10+11+12+13+14+15+16+17+18+19+20+21+22+23 231=16+17+18+19+20+21+22+23+24+25+26 231=30+31+32+33+34+35+36 231=36+37+38+39+40+41 231=76+77+78 231 =115+116</span></div><div><span face="system-ui, -apple-system, BlinkMacSystemFont, Segoe UI, Roboto, Ubuntu, Helvetica Neue, sans-serif" style="color: #14171a;"><span style="font-size: 15px; white-space: pre-wrap;">*Hansreudi Widmer</span></span></div><div><div class="css-1dbjc4n r-1awozwy r-18kxxzh r-zso239" style="-webkit-box-align: center; -webkit-box-direction: normal; -webkit-box-flex: 0; -webkit-box-orient: vertical; align-items: center; background-color: #f5f8fa; border: 0px solid black; box-sizing: border-box; display: flex; flex-direction: column; flex: 0 0 49px; font-size: 15px; margin: 0px 10px 0px 0px; min-height: 0px; min-width: 0px; padding: 0px; position: relative; z-index: 0;"><div class="css-1dbjc4n r-18kxxzh r-1wbh5a2 r-13qz1uu" style="-webkit-box-align: stretch; -webkit-box-direction: normal; -webkit-box-flex: 0; -webkit-box-orient: vertical; align-items: stretch; border: 0px solid black; box-sizing: border-box; display: flex; flex-direction: column; flex: 0 1 auto; margin: 0px; min-height: 0px; min-width: 0px; padding: 0px; position: relative; width: 49px; z-index: 0;"><div class="css-1dbjc4n r-1wbh5a2 r-dnmrzs" style="-webkit-box-align: stretch; -webkit-box-direction: normal; -webkit-box-orient: vertical; align-items: stretch; border: 0px solid black; box-sizing: border-box; display: flex; flex-basis: auto; flex-direction: column; flex-shrink: 1; margin: 0px; max-width: 100%; min-height: 0px; min-width: 0px; padding: 0px; position: relative; z-index: 0;"><br /></div></div></div><hr /><b>The 232nd Day of the Year</b><br />232 is the maximum number of regions that the plane can be divided into with 21 lines (how many of the regions would be of infinite area?)<br /><br />232 is a palindrome in base ten, but no other base from 2-9 <br /><br />There are 232 bracelets possible with 8 beads of one color and seven of another.<div><br /></div><div> And from Derek Orr<br />232 is sum of the cubes of the factorials of its digits,<br />232 = (2!)<sup>3</sup> + (3!)<sup>3</sup> + (2!)<sup>3</sup><br /><br />and 232 the sum of the first 11 Fibonacci numbers. 232 = 1+1+2+3+5+8+13+21+34+55+89 <br /><br />If you add up all the proper divisors of a number, n, they can be less than n, as 231 is(deficient), equal to n (perfect, like 6 or 28) or abundant. 12 is the smallest abundant number. Nicomachus wrote only of even numbers because he thought all odd numbers were deficient, but he was wrong. The <b>232</b>nd abundant number is odd, 945. <br /><br />If you raise 232 to the power of the product of its digits, and then add the sum of its digits, you get a prime. The only other known number with this property is 187. *Prime Curios<br /><br /><br />232 /4 = 58, so 59^2 - 57^2 = 232. 232 / 8 = 29 so 31^2-27^2 = 232.<br /><br /> Because 58 is a sum of two squares, 232 is also. 58 =7^2 + 3^2, 232 = 14^2 + 6^2. Students might use this to find the sum of two squares for 522 = 9 x 58. <br /><br />232 is another balanced binary, with four of each ones and zeros, and always the number of ones is greater than, or equal to, the number of zeros.<br /><br />232 is one less than a Fibonacci Prime. <br /><hr /><b>The 233rd Day of the Year</b><br />233 is the only three digit prime that is also a Fibonacci number. It is the only known Fibonacci prime whose digits are all Fibonacci primes, and the sum of its digits is also a Fibonacci prime. *Prime Curios <br /><br />233 is also the last day of the year that is the sum of the squares of consecutive Fibonacci numbers. (A pretty mathematical fact for the day: the sum of the squares of two consecutive Fibonacci numbers is always a Fibonacci number, Students can show that the converse is not true.)<br /><br />There are exactly 233 <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Maximal_planar_graph" title="Maximal planar graph">maximal planar graphs</a> with ten vertices,<sup class="reference" id="cite_ref-7"><a href="https://en.wikipedia.org/wiki/233_%28number%29#cite_note-7"></a></sup> and 233 connected <a href="https://en.wikipedia.org/wiki/Topological_space" title="Topological space">topological spaces</a> with four points.<br /><br />The ratio of 233/144, two Fibonacci numbers is a good approximation to Phi, the Golden Ratio. <br /><br />233 is the smallest prime factor of |(2^{29} - 1 \) *Prime Curios <br /><br />If you start the "see and say" sequence with 233 (a prime), you get 1223, another prime. And if you do it again with 1233, you get another. *Prime Curios<br /><br />233 is the smallest magic constant of a 5x5 prime Magic square. It uses all the odd primes below 100. *Prime Curios <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tMh0XKbvoi4/XwpO29bn5PI/AAAAAAAAM6A/BumKeLJ-f9Akn4iboy8JHh4JCHbyEUfpgCLcBGAsYHQ/s1600/5x5%2Bprime%2Bmagic%2Bsquare.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="292" data-original-width="226" height="320" src="https://1.bp.blogspot.com/-tMh0XKbvoi4/XwpO29bn5PI/AAAAAAAAM6A/BumKeLJ-f9Akn4iboy8JHh4JCHbyEUfpgCLcBGAsYHQ/s320/5x5%2Bprime%2Bmagic%2Bsquare.jpg" width="248" /></a></div><br /><br />233 is the sum of the squares of the first four semi-primes, 4^2 + 6^2 + 9^2 + 10^2.<br /><br />233= 117^2 - 116^2 <br /><br />233 is the sum of eleven consecutive primes, 5 + 7 + 11 + ... + 41 <br /><br />233 is a palindrome in base 3 (22122)<br /><br />233 in Roman numerals uses 2 C's, 3 X's, and 3 I's. One of the Roman Numeral displays that you can find digital root by counting the number of symbols<br /><hr /><b>The 234th Day of the Year</b><br />234 begins a string of eight consecutive digits which is a prime number, 23456789. All the children need to know why there can not be any ordering of all nine digits , 1 to 9 that is prime. Teach one today. <i><b>Casting out nines</b></i> is as old as Iamblichus, and as new as the youngest kid entering kindegarten. <br /><br />234 is the 55th abundant number. It's proper divisors are {1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117} and their sum is greater than 234, it is 312. There are 86 abundant numbers in a non-leap year. 234=6(39) and all multiples two or greater of a perfect number are abundant.<br /><br /><br />9^2 + 6^2 = 117. Can you see how that tells you that 234 = 3^2 + 15^2??? <br /><br />234 is the number of ways to partition 50 intoat most three parts.<br /><hr /><b>The 235th Day of the Year</b><br />235 is the number of trees with 11 vertices. (Counting the number of unlabeled free trees is still an open problem in math. No closed formula for the number of trees with n vertices <i>up to graph isomorphism</i> is known.)<br /><br />If you build an equilateral triangle with nine matchsticks on each side, then subdivide into additional equilateral triangles, there will be a total of 235 triangles of several different sizes. The image shows the subdivision of a equilateral triangle with three matchsticks on a side. Can you find the thirteen triangles in it? <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-EFmZMTV3H0A/VdErwrNF0YI/AAAAAAAAG0c/Kmxl7WZ08q0/s1600/mathcstick%2Btriangles.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-EFmZMTV3H0A/VdErwrNF0YI/AAAAAAAAG0c/Kmxl7WZ08q0/s320/mathcstick%2Btriangles.png" /></a></div><br /><br />235 is a semi=prime (5 x 47) that is the concatenation of the first three Fibonacci primes<br /><br />235 is a palindrome in base 4 (32234), 7 (4547), and 8 (3538), <br /><br />U-235 is the fissile isotope of uranium used in the first atomic bombs.<br /><br />235, like all numbers ending in 5 that are greater than 25, is the difference of two squares of integers that differ by five, 26^2 - 21^2 = 235. Like all odd numbers, it is the difference of two consecutive squares, 235 = 118^2 - 117^2. <br /><br />235 is the tenth Heptagonal (7-gon) number. n*(5n-3)/2 where n =10<br /><hr /><b>The 236th Day of the Year </b><br />236 is the sum of twelve consecutive primes, 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41<br /><br />And 236 is the number of possible positions in Othello after 2 moves by both players. *Erich Friedman (Students might try to figure out how many possible positions are there in tic-tac-toe after 2 moves by each player. Entice them with this number fact from Cliff Pickover @pickover ) <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-O92SniiD3CQ/V-cNyk5ToMI/AAAAAAAAId8/oPTap7QaAzQRVF6l4w_Vjnu8rJDEIdJtgCLcB/s1600/tic%2Btac%2Btoe%2Bgames.jpe" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="https://3.bp.blogspot.com/-O92SniiD3CQ/V-cNyk5ToMI/AAAAAAAAId8/oPTap7QaAzQRVF6l4w_Vjnu8rJDEIdJtgCLcB/s320/tic%2Btac%2Btoe%2Bgames.jpe" width="320" /></a></div><br /><br />The sum of the divisors of 236, 2+2 + 59 =63, the product of it's digits is 2x3x6 = 36, the reversal of the sum of the divisors.*Prime Curios <br /><br />236^2 + 1 is a prime <br /><br />236 is the average of two consecutive primes 233 and 239. <br /><br />236 is a Happy Number. The sum of the squares of the digits under iteration go to 1. <br /><br />Since 236/4 = 59, 236 = 60^2 - 58^2. <br /><hr /><b>The 237th Day of the Year</b><br />it would be a singularly uninteresting number (3 x 79) except that the room number in the film, "The Shining" was switched from 217 in the novel to 237 for the film? It seems that the Timberline Lodge had a room 217 but no room 237, so the hotel management asked Kubrick to change the room number because they were afraid their guests might not want to stay in room 217 after seeing the film. *Visual Memory.co.uk <br /><br />Derek Orr added, 237 = 44th prime + 44 = 193 + 44 What's the next number that equals the n-th prime + n?<br /><br />237 is the difference of two consecutive squares, like all odd integers, 119^2 - 118^2 = 237, but also 41^2 - 38^2 because 6*38+9 = 237. <br /><br />The 237th square pyramidal number, 4465475, is also a sum of two smaller square pyramidal numbers. There are only four smaller numbers (55, 70, 147, and 226) with the same property. *Wikipedia <br /><br />237 is a lucky number, it remains after all sieves in the Eratosthenes-like Sieve of Stan Ulam. " Start with the natural numbers. Delete every 2nd number, leaving 1 3 5 7 ...; the 2nd number remaining is 3, so delete every 3rd number, leaving 1 3 7 9 13 15 ...; now delete every 7th number, leaving 1 3 7 9 13 ...; now delete every 9th number; etc." *OEIS<br /><br />237 is the last of three consecutive odd numbers that have all prime digits. <br /><br />237 not only has all prime digits, any substring of 2 consecutive digits also form a prime, 23, 37. <br /><hr /><b>The 238th Day of the Year</b><br />The 238th day of the year; 238 is an <a href="https://en.wikipedia.org/wiki/Untouchable_number" target="_blank">untouchable number</a>, The untouchable numbers are those that are not the sum of the proper divisors of any number. 2 and 5 are untouchable, can you find the next one? (four is not untouchable, for example since 1+3=4 and they are the proper divisors of 9) Five is the only known odd untouchable number. <br /><br />238 is also the sum of the first 13 primes, and its digits add up to ........wait for it.... 13 (2+3+8 = 13 and 238 = sum of first 13 primes). Only two more year dates is the sum of the first n primes. <br /><br />2<sup>3</sup>=8 (We are tentatively calling these "power equation numbers") *Derek Orr <br /><br />In base 16 238 is a Repdigit, EE (14 x 16 + 14<br /><br />238 is the number of partitions of 34 into powers of two.And also the same for 35.<br /><br />6 times 238 is between a pair of twin primes. <br /><br /><hr /><b>The 239th Day of the Year</b><br />When expressing 239 as a sum of square numbers, 4 squares are required, which is the maximum that any integer can require; it is the largest number that needs the maximum number (9) of positive cubes. 239 = 5^3 + 3^3 + 3^3 + 3^3 + 2^3 + 2^3 + 2^3 + 2^3 + 1^3. (Only one other number requires nine cubes, can you find it?) <br /><br />and a hundred years (+/-) ago (many people included 1 as a prime then; <a href="http://pballew.blogspot.com/2013/06/one-is-prime-if-we-wish-it-to-be.html" target="_blank">see more</a>) 239 would have been a prime that is the sum of the first 14 primes; 239 = 1+2+3+5+7+11+...+37+41 *Derek Orr <br /><br />The sum of the odd numbers from 1 to 239 is equal to the sum of the odd numbers from 239 to 337.<br /><br />239 appears in one of the earliest known geometrically converging formulas for computing Pi: Pi/4 = 4 arctan(1/5) - arctan(1/239) *.archimedes-lab.org <br /><br />239 is the 52nd prime, and the smaller of a pair of twin primes with 241.<br /><br />For any numbern, greater than 239, the largest factor of n^2 + 1 is at least 17. *Prime Curios<br /><br />If you factor 1234567654321, the smallest prime factor is 239. *Prime Curios<br /><br />Arctan(1/239) in degrees begins 0.239..., and this is the only positive integer for which this is true. *Prime Curios <br /><br />The 52nd prime number (239) has a Collatz trajectory length of 52. It is also the smallest integer greater than 1 that has the same Collatz trajectory length as its square (57121) *Prime Cuiros<br /><br />There are 239 primes less than 1500.<br /><br />239/169 is a convergent of the continued fraction of the square root of 2, so is is a solution to the Pell Equation 239^2 = 2 · 169^2 − 1.<br /><br />The only solutions of the Diophantine equation y2 + 1 = 2x4 in positive integers are (x, y) = (1, 1) or (13, 239).*Wikipedia <br /><hr /><b>The 240th Day of the Year</b><br />240 has more divisors (20 of them) than any previous number. What would be the next number that has more? (Yet the number before it, and after it are both prime!)<br /><br /> 240 can be expessed as a 3x3 magic square by multiplying each of the basic sauares by 16, or with consecutive numbers from 80 to 89.<br /><br />Because 240 is divisible by 20 (the number of its divisors, it is called refactorable. And since it is divisible by the sum of its digits, it is called a Harshad or Joy-Giver Number. <br /><br />240 is the smallest number expressible as the sum of consecutive primes in three ways, *Prime Curios (113+127, 53+59+61+67, 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43.) <br /><br />240 is the product of the first 6 Fibonacci numbers, 240 = 1*1*2*3*5*8 *Derek Orr<br />These are often called Fibonacci factorials or fibonorials. 240 would be \(6!_F\), also called the Fibonacci factorial<br /><br />The Kissing Number, the number of spheres that can be placed around a central sphere so that they all are touching it, for the eighth dimension is 240. Beyond the fourth dimension, only the eighth and twenty-fourth are known exactly. The 24th dimension is the highest dimension for which the exact "kissing number", is known. For the 24th dimension, the "kissing number is 196,560. <br /><br />For those of you who remember Piet Hein's Soma Cube puzzle ("CAn you solve it? Soma Can, Soma Can't!") there are 240 different solutions. <br /><br />The Datsun 240 Z was called the "Fairlady Z" in Japan. My son owned one (very used, and maintained by himself and his brotherfrom two more scraped versions) while he was in college. <br /><br />Because it has so many factors of two, it is expressible as the difference of two squares in several ways, \(240= 61^2 - 59^2 = 32^2 - 28^2 =19^2 - 11^2\) <br /><hr /></div></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-53033951980068306392020-06-23T16:00:00.010-07:002021-05-25T19:06:23.156-07:00Number Facts for Every Year Date - 181-210<b>The 181st Day of the Year</b><br /><div class="separator" style="clear: both; text-align: center;"><span style="font-family: inherit; font-size: small;"><span br="" curios="" prime=""><span style="font-family: inherit;"><span style="font-family: inherit;"><a href="https://1.bp.blogspot.com/-9BmU58FrM8Q/XkHow-BJ9oI/AAAAAAAALMg/wemCs8xid58LFPOzyD9njAXMeMSuONkQQCLcBGAsYHQ/s1600/1961%2BMad%2BUpside%2Bdown%2Byear.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="382" data-original-width="297" height="200" src="https://1.bp.blogspot.com/-9BmU58FrM8Q/XkHow-BJ9oI/AAAAAAAALMg/wemCs8xid58LFPOzyD9njAXMeMSuONkQQCLcBGAsYHQ/s200/1961%2BMad%2BUpside%2Bdown%2Byear.jpg" width="155" /></a></span></span></span></span></div><br /><br />181 is the 42nd prime number, and the largest year day which is a strobogrammatic prime, and the third strobogrammatic prime. It is also the last strobogrammatic number of the year dates..<br /><br />181 is the 9th palindromic prime number. the tenth palindrome prime is not for away, it's the next prime, 191. *Prime Curios adds that this is the smallest pair of consecutive primes that are consecutive, and also the smallest example of two consecutive primes with no prime digits. 181 is also a palindrome when written in base 12(131)<br /><br />111, also a strobogram and is related to 181 in that 111! has 181 digits *Prime Curios<br /><br />If you square 181 and add 7, you get 32768. So what? Well 32768 is 2<sup>15</sup>. STILL not impressed? The only other numbers for which n<sup>2</sup> + 7 is a power of 2 are 1, 3, 5, and 11.... full stop. And to take this beyond the coincidental, if you replace the 7 with any other integer, there will never be more than two solutions. The problem was posed by Ranujan in 1915, who asked were there any others than these five.<br /><br />181 is the sum of 23 consecutive primes, 2+3+5+<sup>...</sup>+79 + 83 =181 *Prime Curios and also the sum of fiv e consecutive primes 29+31+37+41+43= 181<br /><br />*Prime Curios has "The smallest palindromic prime that remains prime through four iterations of the function f(x) = 2x + 5." leaving the suggestion that there are other smaller primes that remain prime through four iteration of f(x) = 2x+5. 7 works three times, but not the fourth. You have to go just a little higher.<br /><br /> and the 181-digit palindromic number made up of all 7's except for the center being 181 (7777...7718177...77777) is a palindromic prime with a palindromic prime decimal length.<br /><br />181 is the both the difference and the sum of consecutive squares:<br />\( 181 = 91^2 – 90^2 = 9^2 + 10^2 \)<div><br /></div><div>181 is also the hypotenuse of a near-isosceles right triangle with the longer side of 180, and the shorter side of 19. <br /><br />181 is the fifth prime (but the first palindromic prime) for which 2^p bisects the gap between two primes, \(2 ^ {181} \pm 165\) are two consecutive primes. *Prime Curios<br /><br />181 is an Emirp with itself, and the sum of three emirps, 31+71+73 = 181.<br /><br />181 is a centered pentagonal number, the sequence of pentagonal numbers begins 1, 6, 16, 31, 51..<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-5go_3J8ZZvk/XsALYKN05SI/AAAAAAAAMcQ/ksaydP4cFLUw2tj8MLCcz8d-tZiEXuOoACLcBGAsYHQ/s1600/centered%2Bpentagonal%2Bnumber.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="98" data-original-width="240" src="https://1.bp.blogspot.com/-5go_3J8ZZvk/XsALYKN05SI/AAAAAAAAMcQ/ksaydP4cFLUw2tj8MLCcz8d-tZiEXuOoACLcBGAsYHQ/s1600/centered%2Bpentagonal%2Bnumber.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Wikipedia</td></tr></tbody></table><br /><br />Every natural number greater than 181 can be written as sum of cubes of the first two primes. (students might be asked to find all examples of numbers less than 181 that can be written in this fashion, such as 35= 2<sup>3</sup> + 3<sup>3</sup>)<br /><br />A good approximation to the number of seconds in one year from Srinavasa Raghava, \( \pi ^{181/12} \)<br /><hr /><b>The 182nd Day of the Year</b><br />there are 182 connected bipartite graphs with 8 vertices. *What's So Special About This Number<br /><br />The 182nd prime (1091) is the smaller of a pair of twin primes (the 40th pair, actually) *Math Year-Round @MathYearRound(Students might convince themselves that it was not necessary to say it was the smaller of the pair.)<br /><br />While literally every small (less than 945) odd number is deficient, 182 is the 91st even number, and only the 48th even number to be deficient. In all then up to 182, there are 182 deficient numbers, and only 43 that are abundant. "The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100 CE)".*Wikipedia<br /><br />Language time: <br /><blockquote>182= 13*14 is called a pronic, promic, or heteromecic and even an oblong number. Pronic Numbers are numbers that are the product of two consecutive integers; 2, 6, 12, 20, ..(doubles of triangular numbers). Pronic seems to be a misspelling of promic, from the Greek promekes, for rectangular, oblate or oblong. Neither pronic nor promic seems to appear in most modern dictionaries. Richard Guy pointed out to the Hyacinthos newsgroup that pronic had been used by Euler in series one, volume fifteen of his Opera, so the mathematical use of the "n" form has a long history. Oblong is from the Latin ob (excessive) + longus (long). The word oblong is also commonly used as an alternate name for a rectangle. In his translation of Euclid's "Elements", Sir Thomas Heath translates the Greek word eteromhkes[hetero mekes - literally "different lengths"] in Book one, Definition 22 as oblong. . "Of Quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right angled but not equilateral...". (note that with this definition, a square is not a subset of rectangles.)</blockquote>As the product of three distinct primes , 182= 2*7*13, it is also a sphenic or wedge number<br /><br />182 is the smallest pronic number (not ending in zero) whose reversal is a prime. <br /><br />182 is a palindrome in base 3 (20202) and a palindrome and repdigit in base 9 (222)<br /><br />The regular polygon with 182 sides, has exterior angles at each vertex of less than 2 degree. Coxeter called all these evenly sided, 2*n, polygons <b>zonagons </b>and said that they could be divided into n(n-1)/2 parallelograms, and in the case of regular polygons, they will all be rhoumbi (but not all identical rhombi), so the 2*91 = 182 sided zonagon will have 91*45=4095 rhombi (too many to make a good image, so here is an Octadecagon with only 36 from the nice people at Wikipedia) (these disections can be done in a multitude of ways, so I just picked a pretty one).<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-ofi7dChwEGo/XsB_GNwok5I/AAAAAAAAMcc/3xGZ_WrIjaU0YFR3HLEmzICKXJREA6w8gCLcBGAsYHQ/s1600/18-gon-dissection.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="162" data-original-width="160" src="https://1.bp.blogspot.com/-ofi7dChwEGo/XsB_GNwok5I/AAAAAAAAMcc/3xGZ_WrIjaU0YFR3HLEmzICKXJREA6w8gCLcBGAsYHQ/s1600/18-gon-dissection.png" /></a></div><br /><hr /><b>The 183rd Day of the Year</b><br />the concatenation of 183 and 184, 183184 is a perfect square. There are no smaller numbers for which the concatenation of two consecutive numbers is square. (Students might seek the next such pair of numbers. They are small enough to be year dates) <br /><br /><a href="https://1.bp.blogspot.com/-iYCaIR-TVAA/XsiIAMNH2tI/AAAAAAAAMfw/rj8pbnfF5383v4Hg_PKVcdo1wpjiLP0uwCLcBGAsYHQ/s1600/toothpick_animated.gif" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="220" data-original-width="220" src="https://1.bp.blogspot.com/-iYCaIR-TVAA/XsiIAMNH2tI/AAAAAAAAMfw/rj8pbnfF5383v4Hg_PKVcdo1wpjiLP0uwCLcBGAsYHQ/s1600/toothpick_animated.gif" /></a>In the toothpick sequence, at the 18th level, there are 183 toothpicks. "<span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">In</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Geometry" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Geometry">geometry</a><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">, the</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><b style="color: #202122; font-family: sans-serif; font-size: 14px;">toothpick sequence</b><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">is a sequence of 2-dimensional patterns which can be formed by repeatedly adding line segments ("toothpicks") to the previous pattern in the sequence.</span><br /><div style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.5em;">The first stage of the design is a single "toothpick", or line segment. Each stage after the first is formed <span style="background-color: transparent;">by taking the previous design and, for every exposed toothpick end, placing another toothpick centered at a right angle on that end"*WIkipedia The sequence begins 1, 3, 7, 11, 15, 23.. it forms a fractal shape. This Wikipedia gif shows the first three steps. </span></div><div style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.5em;"><span style="background-color: transparent;">The image after 89 steps looks like this:</span></div><div style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.5em;"><span style="background-color: transparent;"><br /></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-TblV9JsRtRo/XsiIineDL3I/AAAAAAAAMf0/ub6p5G1yEcsaVYk1utH8r_GYMADztJjlACLcBGAsYHQ/s1600/toothpick%2B89.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="300" data-original-width="300" src="https://1.bp.blogspot.com/-TblV9JsRtRo/XsiIineDL3I/AAAAAAAAMf0/ub6p5G1yEcsaVYk1utH8r_GYMADztJjlACLcBGAsYHQ/s1600/toothpick%2B89.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*WIkipedia</td></tr></tbody></table><div style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.5em;"><span style="background-color: transparent;"><br /></span></div><div style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.5em;"><span style="background-color: transparent;"><br /></span></div><div style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.5em;"><br /></div>The sum of the first 183 primes minus 183 is prime.<br /><br />183 is the eighth of the 12 year-days which are perfect totient numbers. (There are only 57 such numbers under 10<sup>3</sup>). A list of the perfect totient numbers seems to suggest that all of them are multiples of three, but then you get to 4375, the smallest perfect totient number that is not divisible by 3.[a perfect totient number is a number that is the sum of it's iterated totients, that is, the number of integers smaller than, and relatively prime to 183 + the number smaller than and less than that result, + ... down to one, "For example, start with 327. Then φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327 " *Wik<br />If you multiply every digit of a standard 3x3 magic square by 12, you get a 3x3 magic square with a sum of 180, but if you add one more to each entry....<br />49 109 25<br />37 61 84<br />97 13 73<br /><br /><br />183 is the difference of two squares, 32^2 - 29^2, and of course, like every odd number, it is the difference of the squares ot the consecutive numbers that sum to 183, 92^2-91^2= 183 <br /><br />Lagrange proved that every integer is the sum of four or less non-zero squares. 183 is one of the unusual ones that require four. It is the 29th number that requires the full set of three squares. \(183 = 13^2 + 3^2 + 2^2 + 1^1.\)<br /><hr /><b>The 184th Day of the Year </b><br />The 184th day of the year; 184 = 23 * 2<sup>3</sup> (concatenation of the first two primes).<br /><br />The smallest number that can be written as q * p<sup>q</sup> + r * p <sup>r</sup>, where p, q and r are distinct primes (184 = 3 * 2<sup>3</sup> + 5 * 2<sup>5</sup>). *Prime Curios <br /><br />184 is the sum of four consecutive prime numbers 41+43+47+53 = 184<br /><br />184 is a balanced number in binary, with equal numbers of zeros and ones. It is the 14th such number in the year so far.<br /><br />184^2 + 1 =33857 is a prime<br /><br />The concatenation of 183 and 184, 183184 is a perfect square. There are no smaller numbers for which the concatenation of two consecutive numbers is square. (Students might seek the next such pair of numbers. They are small enough to be year dates)<br /><br />On a 5x5 lattice (square grid of dots) there are 184 paths from one corner to the opposite corner touching each lattice point exactly once.<br /><br />184 is a refactorable or Tau number, it's divisible by the count of its divisors. It has eight divisors, including 8. (1, 2, 4, 8, 23, 46, 92, 184).,<br /><br />184 is the difference of two squares 25^2 - 21^2 aand the sum of three squares 12^2 + 6^2 + 2^2 and of four squares, 10^2 + 8^2+4^2 + 2^2<br /><hr /><b>The 185th Day of the Year</b><br />The 185th day of the year; the decimal expansion of the first 185 digits of Euler's constant is prime. *Prime Curios<br /><br />Numbers ending in five, like 185, are always the difference of squares that are five apart, in this case 21^2 - 16^2 =185, and of course, every odd number is the difference of two consecutive squares, 93^2 - 92^2.<br /><br />185 is a semiprime, the product of two distinct primes, 5*37.<br /><br />185 is the sum of two square numbers in two different ways: \( 13^2+ 4^2 \) and \(11^2 + 8^2 \) I'm not sure it is commonly known that this implies that these pairs can be used as the opposite sides of a quadrilateral forcing the diagonals to be perpendicular. (if the sides of a quadrilateral are 13, 11, 4, 8; then the quadrilateral has perpendicular diagonals.)<br /><br />That also means that 185 is the hypotenuse of four Pythagorean Triangles,<br />(60, 75, 185) (111, 148, 1854)(57, 176, 185) (104,153,185)<br /><br />185 =the sum of five squares, 100+64+16+4+1<br /><br />185 is a palindrome in base 6(505) 5*6^2 + 5.<br /><hr /><b>The 186th Day of the Year</b><br />There are 186 days between the Spring and Fall Equinox, which is well over 1/2 a year. The reason, we are on the wrong side of the Earth's Elliptic orbit and have to travel a greater distance. From Fall to Spring takes only 179 days. (there is of course, an extra quarter of a day in there somewhere.) <br /><br />186 is the product of the first four primes less; the product of the first four positive integers, 7# - 4! (7 x 5 x 3 x 2 - 4 x 3 x 2 x 1 = 186) . *Prime Curios Students might not have seen the p# symbol, it represents the Primorial, the product of all the primes from p down to 2. <br /><br />186 is the sum of consecutive primes, 186 = = 89 + 97, <br /><br />186 is a sphenic (wedge) number, product of 3 distinct primes: 186 = 2*3*31<br /><br />Another number with a nice palindrome expressions, 3*3*3 + 3+13 + 31*3 + 3*3*3 (easy as 1,2,3, but without the 2)<br /><br />and from Jim Wilder @wilderlab An equation for July 4th: 7⁴ = 2401 (2 + 4 + 0 + 1 )<sup>4</sup> And a follow up from World Observer@WKryst2011 points out that there are only two other such year dates. (student's should find both)<br /><br />186 is a palindrome in base 5(1221), and in base 8(272) <br /><hr /><b>The 187th Day of the Year </b><br />187^(1*8*7)+1+8+7 is prime. There are only two such (non-zero) numbers. Students might search for the other.<br /><br />The 187th prime is 1117. 11*17 = 187 <br /><br />187² and 187³ don't have 1, 7, or 8. *Math Year-Round @MathYearRound <br /><br />With 187 people in a room, there's a 50% chance that 4 share the same birthday *Derek Orr<br /><br />187 is the sum of three consecutive primes 59, 61, and 67 ; but also the sum of nine consecutive primes starting at 7, and ending at 37.<br /><br />Sesame Street Muppet Count von Count's favorite number, 34969, is 187x187 (1872), speculated in a BBC mathematics program to be a reference between Count Dracula and the Californian murder code (sect 187 of the California Criminal code, sometimes used as a slang word for murder).Sesame Street Muppet Count von Count's favorite number, 34969, is 187x187 (1872), speculated in a BBC mathematics program to be a reference between Count Dracula and the Californian murder code.<br /><br />187 is hexdecimal (base 16) is a tiny number, as small as a BB (there is a joke hidden in there somewhere.)<br /><br />Every odd number is the difference of the squares of two numbers that differ by one and sum to p, in this case 197 = 99^2-98^2,but 197 is also the difference of two other squares, 14^2 - 3^2 <br /><br />Euler's Pentagonal number theorem is important in the computation of the number of partitions of a number. They alter the Pentagonal number forumla \(p_n = \frac{3n^2 - n}{2}\) which generates the Pentagonal numbers when n is a positive integer, by including after each positive integer, it's opposite. The function values at each of these successive terms become the exponents of the variable in his Pentagonal theorem. I mention this here, of course, because the exponenet of the 22nd term is 187. <br /><hr /><b>The 188th Day of the Year</b><br />188 is the sum of six distinct squares. 1 + 4 + 9 + 25 + 49 + 100 = 188. Any larger number can be formed with no more than five distinct squares. <br /><br />188 is the largest known even number that can be expressed as the sum of two (distinct) primes in exactly five ways. *Prime Curios <i>Students might seek smaller numbers that can be so expressed.</i>.<br /><br />Neither 188<sup>2</sup> nor 188<sup>3</sup> contain a one or an eight. *@Derektionary <br /><br />There are 188 11 bead necklaces using two colors, if the necklace can not be turned over.<br /><br />188 is a Happy number: trajectory under iteration of sum of squares of digits map to 1.<br /><br />The largest known even number that can be expressed as the sum of two (distinct) primes in exactly five ways. *Pimre Curios<br /><br />188 is a product of 4 times a number (47). Any such number is the difference of two squares, one of which is the square of one more than the number n/4, and one of which is the square of one less. 48^2-46^2 = 188 <br /><br />The immortal Casey Jones of country music ballads was a real guy (and born in the town of Cacey in Fulton County, Ky) and on April 30, 1900 he took off from Jackson, Tennessee bound for Canton, Mississippi on the Cannonball, but was killed in a dark foggy night when a stranded train was on his rail in Vaughn, Mississippi. His skilled driving saved his passengers, but his life ended at mile number<b> 188</b> of his final drive. <br /><hr /><b>The 189th Day of the Year</b><br />the product of the primes in a prime quadruplet always end in 189, except for the very first quadruplet 3x5x7x11.(<i>A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p, p+2, p+6, p+8} you can see some of the smaller ones listed<a href="http://en.wikipedia.org/wiki/Prime_quadruplet" target="_blank"> here</a> </i> <br /><br />There are 14 prime years in the 21st Century (2017 will be the third), but the 189th century would be the first to contain as few as five prime years (18803, 18839, 18859, 18869 and 18899). <br /><br />Narayana, an Indian mathematician in the 14th century, came up with an interesting Fibonacci-like series: A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. How many cows and calves are there altogether after n years? For the 15th year, the total is 189. (How many mature and immature?) <br /><br />2357 is a prime number. 23357 is also prime. 233357 is also prime but 2333357 is not, and then 23333357 is; and yes, this is somehow related to the number 189. I came across a sequence on <a href="https://oeis.org/A270831" target="_blank">OEIS</a> which gave "Numbers k such that (7*10^k + 71)/3 is prime." Like you may have, I wondered, "Why would someone search for primes of so unusual a sequence?" Well, if you take those prime numbers, and subtract 2 from them, you get the number of threes that when placed between the digit 2 and the digits 57, will produce a prime. So I can inform you today that not only is (7*10<sup>189</sup> +71 )/3 a prime number, but that prime number is a 2 followed by 187 threes followed by 57. <i>And you thought 189 was just some hum-drum number!!!!!</i><br /><br />189 = 13^2 + 4^2 + 2^2 and<br /><br />189 = 1+89 + 98 + 1 a palindrome using only the digits of the number twice each.<br /><br />\(189 = 95^2 - 94^2 \) and of course, as previously mentioned, it is 6^3 - 3^3; and \(189 =5^3 + 4^3 = 15^2 - 6^2= 17^2-10^2\)<br /><hr /><b>The 190th Day of the Year</b><br />The 190th day of the year; 190 is the largest number with only distinct prime Roman numeral palindrome factors that is a Roman numeral palindrome (190 = CXC = II * V * XIX). *Prime Curios <br /><br />190 is also a palindrome in base 4(2332) 190 is a Harshad or Niven number divisible by the sum of its digits. In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-Harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. <br /><br />190 is the 19th triangular number, the sum of the first 19 integers. A nice problem relating triangular numbers to magic squares was asked in 1941 in the American Mathematical Monthly, posed by Royal Vale Heath, widely known for creating ingenious mathematical puzzles: "What is the smallest value of n for which the n2 triangular numbers 0, 1, 3, 6, 10, . . . n2(n2 – 1)/2 can be arranged to form a magic square?" An explanation, and answer is in <a href="http://mathtourist.blogspot.com/2007/11/triangular-numbers-and-magic-squares.html" target="_blank">this blog by Ivars Peterson</a><br /><br />190 is a Happy Number. Summing the squares of the digits, and iterating, you eventually arrive at 1. It takes only four iterations.<br /><br />190 = 121 + 49 + 16 +4 = 100+81+9<br /><hr /><b>The 191st day of the Year</b><br />191 is a palindromic prime and when it is doubled and one is added to this result, the resulting number is yet another palindromic prime. <i>(Students might consider why 11 is the only palindromic prime with an even number of digits.)</i><br /><br />Derek Orr noticed that 199*n+(n-1) is a palindrome (not prime) for several other values of n, collect the whole set.<br /><br />By adding up the values of the common US coins, one obtains 191 ¢ (silver dollar + half dollar + quarter + dime + nickel + penny) *From Number Gossip (This ignores the once minted 5 mil, or half-cent coin and the briefly lived 2 cent coins) <i>Canadians would have a larger sum of coins since <span face=""roboto" , "arial" , sans-serif" style="background-color: white; color: #222222; font-size: 13px;">Canada has had a $1 </span><b style="background-color: white; color: #222222; font-family: Roboto, arial, sans-serif; font-size: 13px;">coin</b><span face=""roboto" , "arial" , sans-serif" style="background-color: white; color: #222222; font-size: 13px;"> (The Loonie) since 1987 and a </span><b style="background-color: white; color: #222222; font-family: Roboto, arial, sans-serif; font-size: 13px;">$2 coin</b><span style="background-color: white;"><span face=""roboto" , "arial" , sans-serif" style="color: #222222;"><span style="font-size: 13px;"> (The Toonie) for about 10 years. I think the Canadian total would be 341 (no half dollar) so maybe we can squeeze them in by the end of the year.</span></span></span></i><br /><br />191 is the smallest palindromic prime p such that neither 6p - 1 nor 6p + 1 is prime. Also, The smallest multidigit palindromic prime that yields a palindrome when multiplied by the next prime: 191 * 193 = 36863. *Prime Curios<br /><br />191 is in the lazy caterer sequence, since 19 slices of a pie can produce 191 pieces. 19 is related to 191 in another way, One Hundred Nighty-One, count the digits.<br /><br />191 is formed by three square numbers, 191^2 = 36481, a square (4) with a square in front of it (36) and behind it (81)<br /><br />191 is a palindrome in base 6 (515)<br /><br />191 is the first prime in a prime quadruplet, 191, 193, 197, 199. The sum of their digits are also prime 11, 13, 17, and 19.<br /><hr /><b>The 192nd Day of the Year</b><br />The 192nd day of the year; 192 is the smallest number that together with its double and triple contain every digit from 1-9 exactly once. 192 + 384 = 576. There are three other values of n so that n, 2n, and 3n contain each non-zero digit exactly once. Can you find them?<br /><br />192 is the sum of ten consecutive primes (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37)<br /><br />192 is the number of edges on a 6th dimension hypercube, it is the last day of the year which is the number of edges of a hypercube.<br /><br />192 is a Happy number, summing the s1uare of its digits and iterating leads to 1 in only three iterations. Its also a Hashard (Joy-giver) number, divisible by the sum of it's digits. <br /><br />Diophantus probably knew, and Lagrange proved, that every positive integer can be written as a sum of four perfect squares. Jacobi] proved the stronger result that the number of ways in which a positive integer can be so written equals 8 times the sum of its divisors that are not multiples of 4. Use this theorem to prove that there are 192 ways to express 14 as a sum of four squares. <br /><br />Because 192 is divisible by four, it is the difference of two squares n^2 and (n+2)^2, 49^2 - 47^2; because it is divisible by 8, it is the difference of two squares that n^2 + (n+4)^2; 26^2 - 22^2. But don't stop there, because it is divisible by 16, it is the difference of two squares n^2 and (n+6)^2; 19^2 - 13^2 , and YES, it it also divisible by 32, so one more time, 16^2-8^2.<br /><br />192 = 3*4^3 and also one more than 3 * 8^2 <br /><br />192 is called a practical number, because you can make a subset of its divisors to sum to any number less than 192. All perfect numbers are practical, as are all Harmonic divisor numbers. <br />\(n*2^{n-1} \) gives the number of edges (segments) in a n-dimensional cube, and in the 6th dimension, called a hexeract, there are 192 edges, 6*2<sup>5</sup><br /> <hr /><b>The 193rd day of the Year</b><br />193 = 12^2 + 7^2<br />193 is also 97^2 - 96^2<br /><br />193 is one of a twin prime pair and is the sum of products of the first three twin primes pairs: 3*5 + 5*7 + 11*13 = 193. *Prime Curios The square of 193 (37249) concatenated with its reverse (which is a prime) results in a palindrome (3724994273) that is the product of 2 palindromes, one non-prime (1001) and one prime (3721273). *Prime Curios <br /><br />The product of the first four primes, 2*3*5*7 = 210, their sum is 17, the product minus the difference is 193. <br /><br />193 is a happy number, if you sum the square of the digits, and do the same with each answer, you will eventually wind up with 1. <br /><br />Like all odd numbers, it is the difference of two consecutive squares, 97^2- 96^2 and 96+97 = 193. <br /><br />193 is a palindrome in base 12 (121) so 193 = 12^2 + 2*12 + 1<br /><br />193 is one more than 3*4^3 and also one more than 3 * 8^2 193 is the smallest prime whose fifth power contains all digits from 1 to 9. <br /><br /><i>(I also like 193/71 is the closest ratio of two primes less than 2000 to the number e.</i>) <br /><br /><hr /><b>The 194th Day of the Year</b><br />194<sup>4</sup>+1 = 1,416,468,497 is prime *Prime Curios 194 is also the smallest number that can be written as the sum of 3 squares (not all unique) in five ways. (There is a slightly larger number that is expressible as the sum of 3 unique squares in five ways. ) <br /><br />194 = 13^2 + 5^2<br /><br />194 is the smallest Markov number that is neither a Fibonacci number or a Pell Number, Markov nuimbers are any of the x, y, z, such that x^2 + y^2 + z^2 = 3xyz, and in this case they are 5, 13, 194. ( I admit I have some work to do to figure out what makes these triples special and the chains on which they are formed. )<br /><br />194 is a palindrome in base 3, (21012), and in base 6 it is (1234) which is cool.<br /><br />194 is the product of the largest and smallest prime less than 100. <br /><br />194 is the sum of three consecutive squares, \( 194 = 7^2 + 8^2 + 9^2 \)<br /><br />194 is the even base of the Largest Heronian triangle with consecutive integer sides that can be year dates. Heronian triangles are triangles that have all three sides and the area as integers. The three sides are 193, 194, 195, and the Area is 16,296 sq units. I have seen these called Super Heronian triangles, but <span style="background-color: white; color: #222222; font-size: 16px;">I call them Sang-Heronian triangles after the earliest study I know about them by Edward Sang of Edinburgh, Scotland in1864</span> Because these bases are always equal, the altitude from that base must also be an integer. And one more biggie... If you construct the altitude to the even base, one side or the other of it will always form a primitive Pythagorean triangle. For each new bigger triangle, it switches sides. In the triangle for this date, the PPT is 95, 193, 195. I won't to write a little more about this than space here allows, so I will link it here then.<br /><hr /><b>The 195th Day of the Year</b><br />Take the basic 3x3 magic square, multiply by 13, and get<br><br /> 52 117 26<br />39 65 91<br />104 13 78<br />A magic square with a constant of 195. 13*15=195 different multiple gives you constants of 15k for any k.<br /><br />195 is a palindrome in binary (11000011) and as you can see, it is a balanced number with exactly the same number of zeros and ones. It is also a palindrome in base 4. Divided the digits into groups of two and convert their value to decimals, 11=3, 00=0, 00=0, and 11=3 so in base four, it is (3003), and in base 8 (303) by taking three digits at a time (right to left) 11, 000, 011<br /><br />If you take the basic 3x3 magic square with digits from 1 to 25, and multiply each term by 3, you get a 5x5 magic square with a magic constant for each row and column of 195<br />51 72 3 24 45<br />69 15 21 42 48<br />12 18 39 60 66<br />30 36 57 63 9<br />33 54 75 6 27<br /><br />if you keep 39 in the middle and replace all numbers in their order by incrementing by one, you get another magic square for 195. (42 will be 40, 36 will be 38, etc.)<br /><br />Add one to each term and you have a magic square for day 200!<br /><br />195 is in the sequence of integers 6n+9 for n = 31 and so 195 = 34^2 = 31^2<br />and because it is in 10n+25 for n=17, 195 = 22^2 - 17^2, and like every odd number, 195 = 98^2 -97^2, , the difference of the squares of two consecutive numbers that add up to 195.<br /><br />195 is the sum of eleven consecutive primes: 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 Students might wonder which numbers can (and cannot) be expressed as the sum of one or more consecutive Primes. <br /><br />195=5^2 + 7^2 + 9^2 It is only the third integer that is the sum of the squares of three consecutive primes.*Prime Curios 5<br />There are lots more ways to find distinct squares that sum to 195, <span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">as it is the smallest number expressed as a sum of distinct </span>squares<span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> in 16 different ways. *Wik</span><br /><br />also, 1*95 = 19*5 Derek Orr tells me there are only four non-trivial 3-digit numbers with this property *@Derektionary<br /><br />a Heronian triangle is a triangle that has side lengths and area that are all integers. There is an almost-equilateral, scalene triangle with one side of 195. The other sides are 194, and 193. Students can find the area using Heron's formula. <br /><hr /><b><span style="font-family: "courier new" , "courier" , monospace;">The 196th Day of the Year</span></b><br /><span style="font-family: "courier new" , "courier" , monospace;">14^2 = 196;</span><br /><span style="font-family: "courier new" , "courier" , monospace;"><br /></span><span style="font-family: "courier new" , "courier" , monospace;">A Lychrel number is a natural number which cannot form a palindromic number through the iterative process of repeatedly reversing its base 10 digits and adding the resulting numbers. 196 is the lowest number conjectured to be a Lychrel number; the process has been carried out for one billion iterations without finding a palindrome, but no one has ever proven that it will never produce one. The number produced on the one billionth iteration had 413,930,770 digits<b>. </b>The name "Lychrel" was coined by Wade VanLandingham—a rough anagram of his girlfriend's name Cheryl. No Lychrel numbers are known, though many numbers are suspected Lychrels, the smallest being 196. (Students might try finding the number of iterations of the process to find a palindrome for various n. 195, for example, takes four iterations : 195 + 591 = 786 786 + 687 = 1473 1473 + 3741 = 5214 5214 + 4125 = 9339) DO not try the numbers 89 or 98. Harry J Saal used a computer to repeatedly iterate this process and finally did come up with a palindrome, the number 8,813,200,023,188 on the 24th iteration. </span><br /><span style="font-family: "courier new" , "courier" , monospace;"><br /></span><span style="font-family: "courier new" , "courier" , monospace;">Jim Wilder noticed that 14<sup>2</sup> =196 and 13<sup>2</sup>=169... are there other squares of consecutive numbers that share the same digits? </span><br /><span style="font-family: "courier new" , "courier" , monospace;"><br /></span><span style="font-family: "courier new" , "courier" , monospace;">196 is the aliquot sum of </span><span style="background-color: white; color: #202122; font-size: 14px;"><span style="font-family: "courier new" , "courier" , monospace;">140, 176 and 386</span></span><br /><span style="background-color: white; color: #202122; font-size: 14px;"><span style="font-family: "courier new" , "courier" , monospace;"><br /></span></span><span style="background-color: white; color: #202122; font-size: 14px;"><span style="font-family: "courier new" , "courier" , monospace;">196 is a palindrome in base 13, (212) (I never do had much luck working in base 13)</span></span><br /><span style="background-color: white; color: #202122; font-size: 14px;"><span style="font-family: "courier new" , "courier" , monospace;"><br /></span></span><span style="background-color: white; color: #202122;"><span style="font-family: "courier new" , "courier" , monospace;">and a palindromic expression of 196 useing only its digits, 19 + 16 + 9 + 61 + 91 OR 96 +1 + 9 + 11 + 9 + 1 + 69 </span></span><br /><span style="font-family: "courier new" , "courier" , monospace;">A number is said to be square-full if for every prime, p, that divides it, p<sup>2</sup> also divides it. 196 is such a number, 196 = 2^2 + 7^2 Are there cube-full numbers? (of course there are, but what are they? 8 would be, as would any cube, I guess smallest with more than one is 6^3 = 216, coming up soon)</span><br /><span style="font-family: "courier new" , "courier" , monospace;"><br /></span>Because 196/4 = 49, it must be the difference of two squares, 50^2-48^2.,</div><div><br /></div><div>If you take an integer,<i> almost</i> any integer it seems, and add the number to its reverse, then continue this process with each succeeding sum until you come to a palindromic number, you almost certainly will. 38 takes only one addition, 38 + 83 = 121. 139 takes two, 139 + 931 = 1070 and 1070 + 0701 = 1771. But take a bite out of 196, and you may have to give up before you find it. Computers have gone through thousands of iterations and, to my knowledge, never found a palindrome. </div><div>Any suggestions for why this one number out of all the integers, might not have a palindromic terminal?<br /><hr /><b>The 197th Day of the Year</b><br /><br />197 is the sum of all digits of all two-digit prime numbers: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. It is simple to show that the sum of one-digit primes is 17. Do the sum of the digits of n-digit primes always end in seven? (http://oeis.org/A130817) Or perhaps we ask, are there any others that do?<br /><br />196 was a square-full number, and 197 is a square free number, since it is the 43rd prime<br /><br />197 is the smallest prime number that is the sum of 7 consecutive primes: 17 + 19 + 23 + 29 + 31 + 37 + 41 (student challenge: can there be a prime that is the sum of eight consecutive primes?)<br /><br />197 is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37, and obviously the smallest number to be the sum of twelve primes. (You have to wait 39 days for the next one.<br /><br />197 is the last year date which is a Keith number . The Keith numbers is a digit summing process in which one digit is replaced by the new sum. For example, since 197 is a three digit number, we will add three digits in each iteration. We start with the digits of the number, 1+9+7 = 17. Now we delete the first digit, 1, and add the 9+7 +17 = 33, Again, delete the first of the three numbers, 9 and add 7 + 17 + 33 = 57. Continue the process 17+33+57= 107, and once more for; this one, 33+57+107 = 197, back to the number we started with. Such numbers are called Keith Numbers after Mike Keith, an American mathematician working at Sarnoff created the idea in 1987. There are only about 100 known in base ten, and only 7 that are year dates, 14, 19, 28, 47, 61, 75, and 197. Keith used the term repfigit, and that name is still used by some today. <br /><br />197 = 111 + 9 +77 sum of three repdigits<br /><br />197 = 14^2 + 1^2 and 197 = 99^2-98^2<br /><br /><hr /><b>The 198th Day of the Year;</b><br />198 is a Harshad number, divisible by the sum of its digits. A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The Niven numbers take their name from Ivan M. Niven from a paper delivered at a conference on number theory in 1997. (Students might try to find a pair of consecutive numbers greater than 10 which are harshad numbers) <br /><br />198 nines followed by a one is prime 9999...... 91. *Derek Orr@Derektionary <br /><br />198 is between the twin primes 197 and 199. <br /><br />If you multiply 198 by its reversal, 891, you get 176,418 which is between the twin primes 176,417 and 176,419. Is there another example of this curiosity?<br /><br />198 = 13^2 + 5^2 + 2^2<br /><br /> Palindrome expressions for 198 = 2 x 72 + 27 x 2 = 3 x 15 + 51 x 3 = 55 + 88+ 55<br /><br />19 is the 8th prime number, and if you concatenate them, 198 is the (19*8= 152nd) composite number. <br /><br />The difference between any two Emirp pairs is divisible by 198 *Prime Curios<br /><br />198 is called a practical number because every number from 1 to 197 can be written as sums of divisors of 198. <br /><br /><hr /><b>The 199th Day of the Year</b><br />199 is prime (in fact, all three permutations of the number are prime) and is the sum of three consecutive primes: 61 + 67 + 71, and of five consecutive primes: 31 + 37 + 41 + 43 + 47. (Suddenly struck me I don't know what is the smallest prime that is the sum of consecutive primes in more than one way!)(And now I know, it is 41)<br /><br />199 is the sum of the digits of all the three-digit palindromic primes. *Prime Curios <br><br><br />199 is the smallest number with an additive persistence of 3. (iterate the sum of the digits. The number of additions required to obtain a single digit from a number n is called the additive persistence of n, and the digit obtained is called the digital root of n. ) 1+9+9 =19, 1+9=10, 1+0 = 1. so the additive persistence is 3 and the digital root is 1.<br /><br />I like "almost constants". For the 199th day, \( ( \frac{\sqrt{5} +1}{2})^{11}= 199.0050249987406414902082… \)<br /><br />199 is the last year day that is part of a prime quadruplet, (191, 193, 197, 199) <br /><br />199 is the smallest number that has an additive persistence of 3, 1+9+9 = 19; 1+9 =10; 1+0=3 *Prime Curios<br /><br />199 = 100^2-99^2<br /><br />199 as a palindrome of its own digits, 99+1+99=199= 9*9+9*1+9+1+9+1*9+9*9<br /><br />199 is a permutable prime, and 919 and 991 are both prime <br /><br />199 is the first prime number in a sequence of 10 consecutive prime numbers with common difference 210 (tao and green 2008; see R.Taschner "Die Farben der Quadratzahlen" p. 147)the ten primes are 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, and 2089.<br /><br />The next prime after 199 is 211. If they are concatenated in either order, they form a prime, 199211 and 211199 are both prime. *Prime Curios<br /><br />199 is the smallest emirp that is also an invertible prime, it's 180 degree rotation produces the prime 661. *Prime Curios<br /><br />199, 211, and 223 are the smallest triple of primes of the form n, n+12 and n+24, and it is the only triple less than 1000. *Prime Curios <br /><hr /><b>The 200th Day of the Year</b><br />200=14^2+2^2 and of course 10^2+10^2.<br /><br />200=51^2-49^2=27^2-23^2<br /><br />200 is the smallest unprimeable number - it can not be turned into a prime number by changing just one of its digits to any other digit. (What would be the next one? What is the smallest odd unprimeable number? ) <br /><br />Sum of first 200 primes divides product of first 200 primes. (How often is this property true of integers?) *Math Year-Round @MathYearRound<br /><br />The smallest discernible movement of a computer mouse—equal to 1/200th of an inch—is called a MICKEY. Haggard Hawks @HaggardHawks The actual measure depends on the equipment of course, so, like the meter, it will have to be adjusted from time to time.<br /><br />A 5x5 magic square for 200 is easy. Take the standard 5x5 square using 1-25, multiply all entries by 3x+1. ,BR.,BR.<br /><br />51 72 4 25 46 <br><br />70 16 22 43 49<br><br />13 19 40 61 67<br><br />31 37 58 64 10<br><br />34 55 76 7 28<br><br /><br><br><br />And don't forget, When you pass GO, collect $200 (except the lucky Birts, who get 200 Lb.<br><br><br /><br />It is not a palindrome in any base 2-10, but it is in Roman numerals CC. <br><br />Which reminds us that 200 is the sum of two squares, 10^2 + 10^2, but also 14^2 + 4^2, <br>And as the difference of two squares, 200 = 51^2 - 49^2; 27^2 - 23^2; 15^2 - 5^2<br /><br /><hr /><b>The 201 Day of the Year</b><br />201 is a harshad number... A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The Niven numbers take their name from Ivan M. Niven from a paper delivered at a conference on number theory in 1997. (Can you find the string of three consecutive Harshad numbers smaller than 201?)<br /><br />201 is also a lucky number, a number that survives from the sieve process created about 1955 by Stanislaw Ulam, the great Polish mathematician who coinvented the H-bomb and was the father of cellular automata theory. Students who are familiar with the way the Sieve of Erathosthenes produces the primes may wish to compare the lucky numbers produced by this sieve. "Start wtih the odd numbers.The first odd number >1 is 3, so strike out every third number from the list (crossing out the 5, 11,17 etc): 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, .... The numbers that remain are the so called "lucky numbers". Look for similarities to the primes. *Martin Gardner, Mathworld<br /><br />201= 101^2-100^2 = 35^2-32^2<br /><br />201 has a largest factor of 67, so a palindrome of 201 with only 6 and 7, 6 x 7 + 7 x 6 + 7 + 6 + 7 + 6 + 7 + 6 x 7 + 7 x 6 = 201<br /><br />201 is a Joy-Giver or Harshad number, divisible by the sum of its digits<br /><br />201 is the the difference of two squares, 201 = 101^2 - 100^2=35^2 - 32^2, and just to break the rules, it's also 22.6^2 - 17.6^2<br /><br />And Wikipedia tells me that Star Trek had an episode with the title, 11001001, which is 201 in binary. <br /><br />A 3x3 magic square with a magic constant for each row of 201 can be created by taking the basic 1-9 digits and replacing them with 13n+2<br /><br />54 119 28<br />41 67 93<br />106 15 80<br /><hr /><b>The 202nd Day of the Year</b>in an alphabetical listing of the first one-thousand numbers, 202 is last. <br /><br />202<sup>293</sup> begins with the digits 293 and 293<sup>202</sup> begins with the digits 202. *jim wilder @wilderlab<br /> To get a fill of how rare it would be for this kind of reversal, students might search for just numbers that work only one way. For example 6^13 starts with a 13, but the reversal doesn't work. Sounds like a good idea for a computer search; if you find more, send me a note.<br /><br />There are 46 palindromes in the 365 (or 366) days of the year, 202 is the 30th of these.<br /><br />(2+3+5+7)<sup>2</sup> -(2 <sup>2</sup> + 3<sup>2</sup> + 5<sup>2</sup> + 7<sup>2</sup>) =202 *Prime Curios <br /><br />If your digital clock uses a seven digit display, such as many microwaves, stoves, and small alarm clocks, then 202 is a strobogrammatic number, turn the clock (not the stove, please) over and it reads the same. <br /><br />There are exactly 202 partitions of 32 (2^5) into smaller powers of two *Wikipedia <br /><br />202 in Roman numerals and read in English sounds like a "Yes" in both Spanish and Nautical, "Si, Si, Aye Aye" <br /><br />202 = 11^2 + 9^2 <br /><hr /><b>The 203rd Day of the Year</b><br /><br /><span style="font-family: "courier new" , "courier" , monospace;">203 = 18^2-11^2 = 102^2-101^2</span><br /><br /> 203 is the 6th Bell number, i.e. it is the number of partitions of a set of size 6.<br /><br /><a href="https://3.bp.blogspot.com/-F9Y5mu8Kl88/V4kL3dlJK1I/AAAAAAAAIUM/e-cmcDwg6pMbRETt_tRHNyr6S5spL_xLACLcB/s1600/matchstick%2Btriangle.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://3.bp.blogspot.com/-F9Y5mu8Kl88/V4kL3dlJK1I/AAAAAAAAIUM/e-cmcDwg6pMbRETt_tRHNyr6S5spL_xLACLcB/s200/matchstick%2Btriangle.png" width="200" /></a>203^2 + 203^3 + 1 is prime. <br /><br />203 is the number of nondegenerate triangles that can be made from rods of lengths 1,2,3,4,...,11 <br /><br />203 is the number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side length 13 <br /><br />Saw a tweet about July 22 as "Casual Pi Day" at Rimwe@RimweLLC which he told me he found at page of GeorgeTakei. The NCTM uses "Pi Approximation Day" for it's poster <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-yofEJEgzscc/V5JKxKHoLiI/AAAAAAAAIWQ/m9fhsb3oXyodd_O8epeKriCglPbFLMwXgCLcB/s1600/pi%2Bapprx%2Bday%2Bnctm.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="https://1.bp.blogspot.com/-yofEJEgzscc/V5JKxKHoLiI/AAAAAAAAIWQ/m9fhsb3oXyodd_O8epeKriCglPbFLMwXgCLcB/s320/pi%2Bapprx%2Bday%2Bnctm.jpg" width="320" /></a></div><br />203 is a palindrome in base 3 (21112) base 6 (535), and base 8(313),<br /><br /><br />203 is the sum of the squares of five consecutive primes. No smaller such prime exists, and no other day number has this quality.<br /><hr /><b>The 204th Day of the Year</b><br />204 is the eighth tetrahedral number, the sum of the squares from 1 to 8. Answers the question, how many squares are there on an 8x8 checkerboard.<br /><br />204 = 20^2 - 14^2=52^2 - 50^2&<br /><br />204 is the sum of consecutive primes in two different ways: as the sum of a twin prime (101 + 103) and as the sum of six consecutive primes (23 + 29 + 31 + 37 + 41 + 43). (one might wonder what is the smallest number that is the sum of consecutive primes in more than one way... And what is the smallest <i>prime</i> number that is expressible as the sum of consecutive Primes in more than one way?)<br /><br />And a trio from *Derek Orr @MathYearRound : <br />204 = 1²+2²+3²+4²+5²+6²+7²+8².<br />Sum of first 204 primes is prime.<br />100...00099...999 (204 0's and 204 9's) is prime.<br /><br />204^n + b1 is prime when n = 2, 0, 0r 4. *Prime Curios<br /><br />204 is a refactorable number, it is divisible by the count of its divisors, 12<br /><br />It is because it is divisible by 12 that 204 is expressible as (n+3)^2 - (n-3)^2 for n= 17 (204/12=17)<br /><br />204^2 = 41616, a number that is both a square and a triangular number. They are pretty rare as it is only the fourth. And like 204, it is the sum of twin primes also, 41616 = 20807 + 20809. Only 12 and 84 also are the sum of twin primes with a square that is also the sum of twin primes. <br />since I've written twin primes so many times in this entry, its probably a good time to remind you that, although the idea of primes seperated by only a single composite number was known back to antiquity, the term was created around the end of the 19th century by German Mathematician Paul Stackel (in the German, of course, "Primezahlzwilling")<br /><br />On an infinite chess board, a Knight can reach 204 different squares in eight moves. <br /><br />200 is CC in Roman numerals, and 204 is CC in hexadecimal. <br /><hr /><b>The 205th Day of the Year</b><br />there are 205 pairs of twin primes less than ten thousand. *Number Gossip <br /><br />Every number greater than 205 is the sum of distinct primes of the form 6n + 1. *Prime Curios <br /><br />205 is the number of walks of length 5 between any two distinct vertices of the complete graph K_5 <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-orMqTc4Dc9g/W1Iq3vUHU0I/AAAAAAAAJG0/NTcd5vbiQf0NP-GvB91UKvrjmcYyMGEywCLcBGAs/s1600/k5%2Bcomplete%2Bgraph.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="91" data-original-width="102" src="https://2.bp.blogspot.com/-orMqTc4Dc9g/W1Iq3vUHU0I/AAAAAAAAJG0/NTcd5vbiQf0NP-GvB91UKvrjmcYyMGEywCLcBGAs/s1600/k5%2Bcomplete%2Bgraph.png" /></a></div><br /><br />205 = 14^2 + 3^2 ;= 13^2+6^2<br /><br />205 = 103^2 - 102^2 = 23^2 - 18^2 The first is from a property of every odd number, the second is from a property of any number greater than 35 that ends in five.<br /><br />205 = 2 x 41 + 41 + 41 x 2 also 54 + 45 + 7 + 54 + 45. or 99 + 7 + 99 <br /><br />205 is a palindrome in duodecimal (base 12, (151) =1 x 12^2 + 5 x 12 + 1)<div><br /></div><div>205 is the smallest un-permeable odd composite . It can not be turned into a prime by changing only one of its digits. <br /><hr /><b>The 206th Day of the Year</b><br />206 is the lowest positive integer (when written in English) to employ all of the vowels once only. (This seems to require the use "two hundred AND six" which I really dislike. What would be, or is there a, first without this "and"?) (Michael King @processr suggested "5000 fIvE thOUsAnd". <br /><br />206 is Sum of the lengths of the first runs in all permutations of [1, 2, 3, 4, 5] (for example, the first run of the permutation 23541 is three.)<br /><br />206 is the sum of 39, and the 39th prime, 167. <br /><br />There are 206 bones in the typical adult human body. (I suppose all those spineless people folks talk about have well under 200!) <br /><br />206 is the 36th year date for which n^2 + 1 is prime. It's also the 59th day for which n^2 + n + 1 is prime. Am I the only one who thought the second would be more unusual?<br /><br />204, 206 and 208 are all the sum of a square and a cube. 206= 5^3 + 9^2. It seems that there are an infinite numbers of three consecutive integers that are such sums. 126, 127, 128 and 129 is a string of four such sums<br /><br />There are 206 partitions of 26 into four parts<br /><br />The sum of the divisors of 14 and 15 are equal. I mention that here because the next occurrence of such an incident is 206 and 207 which both sum to 312.<br /><br /><hr /><b>The 207th Day of the Year</b><br />207 is the smallest possible sum of primes which are formed using each of the digits 1 through 9 (i.e., 89 + 61 + 43 + 7 + 5 + 2 = 207) *Prime Curios (So how many such sums can there be? And which of such sums are prime?)<br /><br />There are exactly 207 different matchstick graphs with eight edges ( a <b>matchstick graph</b> is a graph that can be drawn in the plane in such a way that its edges are line segments with length one that do not cross each other) Here are a few of them: <br /><div class="separator" style="clear: both; text-align: center;">207 = 16^2 - 7^2 = 104^2 - 103^2 = 36^2-33^2</div><a href="https://1.bp.blogspot.com/-dCaC_CrCIxo/V4u0xwa3GcI/AAAAAAAAIVE/YS6KiRSOAPseV3ZB337r0HNKZ6sGRv3cgCLcB/s1600/matstick%2Bgraphs.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="161" src="https://1.bp.blogspot.com/-dCaC_CrCIxo/V4u0xwa3GcI/AAAAAAAAIVE/YS6KiRSOAPseV3ZB337r0HNKZ6sGRv3cgCLcB/s320/matstick%2Bgraphs.jpg" width="320" /></a><br /><br /><br />207 =3 x 3 x 23, and 3^207 + 3^207 + 23^207 is a prime. Prime Curios<br /><br />207 = 9 times the 9th prime, and it is the 9th year day that is n times the nth prime.<br /><br />207 = 104^2 - 103^2, and because 6 x 33 + 9 = 207, 207=36^2 - 33^2. <br /><br />207 is the 33rd day of the year that are the sum of 4, but no fewer nonzero squares.<br /><hr /><b>The 208th Day of the Year</b><br />208 is the sum of the squares of the first five primes.<br /><br />208 is the number of paths from (0,0) to (7,7) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps. <br /><br />208 is an abundant number, the proper divisors total 226(more than 208)<br /><br />208 = 6^3 - 2^3<br /><br />208 is the sum of a cube and a square, as were 204 and 206. 208 = 4^3 + 12^2<br /><br />208 = 8^2 + 12^2. <br /><br />208 = 53^2 - 51^2 = 17^2 - 9^2 = 28^2 - 24^2<br /><br />(16*10^208-31)/3 is prime, and it has a 5 followed by 206 threes, finished of with 23. It is the largest year date in this sequence. Previous examples include 523, 5323, 53323, and 5333333333333323, for the exponents 1, 2, 3, 4 and 15<br /><br />\(208 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2\), the sum of the first five squares, obviously the smallest number to be the sum of five distinct squares of primes.<div><br /></div><div><hr /><b>The 209th Day of the Year</b><br />209=1<sup>6</sup>+2<sup>5</sup>+3<sup>4</sup>+4<sup>3</sup>+5<sup>2</sup>+6<sup>1</sup>. <br /><br />Also 209 is a "Self number" A self number, Colombian number or Devlali number (after the town where he lived) is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 209, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar. <i>students might want to explore self numbers for patterns </i><br /><i> </i>[<i>The earliest use of Colombian number I can find is by </i><span class="citation journal">B. Recaman (1974). "Problem E2408". <i>Amer. Math. Monthly</i> <b>81</b><i>. Would love to know if there are earlier uses.</i>]</span><br /><br />209 is a Harshad (Joy-giver) number, divisible by the sum of its digits. The smallest with digit sum of 11. H/T John Golden<br /><br />209 is the maximum number of pieces that can be made by cutting an annulus with 19 straight cuts. <br /><br />209= 105^2 - 104^2<br /><br />The curve 42x2 - y2 = 209 contains the 'prime points' (3, 13), (5, 29), (7, 43), and (13, 83). *Prime Curios<br /><br />There is an infinity of pairs x,y where x2 - y2 - xy = 209 for x, y integer *Prime Curios <br /><br />209 is the smallest number with six representations as a sum of three positive squares. These representations are: 209 = 12 + 82 + 122 = 22 + 32 + 142 = 22 + 62 + 132 = 32 + 102 + 102 = 42 + 72 + 122 = 82 + 82 + 92. *Wikipedia<br /><br />209 is one less than a primorial. Euclid's proof of the infinity of the Primes depends on the fact that n#-1 or n#+1 can not be a factor of any of the primes up to n, hence there must be another. 209 is the smallest n#-1 that is not a prime, but the product of prime factors larger than n=7, (11 x 19)<br /><br />209 is a palindrome in base 6 (545) (Base six is called senery, After the sequence, binary, ternery, quaternery, quinery, senery, septenery...and then they jump to octal???)<br /><br />There are 209 partitions of 16 into relatively prime parts. <div><br /></div><div>As far as I know, there are only two three digit numbers so that abc^2 = uvwxyz and uvw+xyz = abc. These are called three digit Kaprekar numbers. 209 is involved in each case. The two numbers are 297^2 = 88209, with 88 + <b>209</b> = 297 and 703^2 = 494209 where 494 + <b>209</b> = 703<br /><hr /><b>The 210th Day of the Year</b><br />210 is the last year day that is a Primorial, 210 = #7 = 7*5*3*2. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.*Wikipedia Of course that means it is the smallest number that is the product of four distinct primes, and the only such year date. <br /><br />(21, 20, 29) and (35, 12, 37) are the two least primitive Pythagorean triangles with different hypotenuses and the same area (=210). Students are challenged to find another pair of such PPTs <br /><br />There are an infinite number of numbers that appear six or more times in Pascal's Arithmetic Triangle, but only three of them; 1, 120, and 210 are year dates.<br /><br />7! hours is 210 days. <br /><br />The Combination of ten things taken four at a time is 210. Patrick Honaker asks is there another such where k is not 1 or n-1. *Prime Curios <br /><br />13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210, the sum of eight consecutive primes<br /><br />210 is the last year date which is both a triangular number and the product of consecutive numbers, 14 x 15. It is also the last to be the product of three consecutive numbers, 5 x 6 X 7. <div><br />Three different ways to make a 3x3 magic square with a magic constant of 210, Take the classic 3x3 and multiply each term by 14,<br />56 126 28<br />42 70 98<br />112 14 84<br /><br />Or with consecutive integers starting at 76<br /><br />69 74 67<br />68 70 72<br />73 66 71 <br /><br />Or maybe with increments of five<br /><br />65 90 55<br />60 70 80<br />85 50 75 <br /><br />The magic is in the middle, all else stems from there. <br /><br />210 is the 20th Triangular number, the sum of the integers from 1 - 20. <br /><br />210 in binary is a balanced number, with the same numbers of ones and zeros, and reading from left to right the zeros never outnumber the ones. <br />The sum of the squares of the divisors of 12, is 210. <br /><hr /></div></div></div></div></div>Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-4490843217324699740.post-8591152031326116282020-05-14T16:47:00.020-07:002021-05-28T08:55:40.048-07:00Number Facts for Every Year Date, 151-180 <b>The 151st Day of the Year</b><br />151 is the 36th prime number, and a Palindromic Prime. Did I ever mention that palindrome is drawn almost directly from an Ancient Greek word that literally means "running back again." First used in English in 1636 in "Camdon's Remains Epitaths". <br /><br />The smallest prime that begins a 3-run of sums of 5 consecutive primes: 151 + 157 + 163 + 167 + 173 = 811; and 811 + 821 + 823 + 827 + 829 = 4111; and 4111 + 4127 + 4129 + 4133 + 4139 = 20639. *Prime Curios... Can you find the smallest 4-run example?<br /><br />151 is also the mean (and median) of the first five three digit palindromic primes, 101, 131, 151, 181, 191 <br /><br />151 is an undulating palindrome in base 3 (12121) <br /><br />Thanks to Derek Orr, who also pointed out that any day in May (in non-leap year) 5/d is such that 5! + d = year day<br /><br />In 1927 Babe Ruth hit 60 Home Runs, a long lasting record. He hit them in 151 games.<br /><br />And from base to torch, Lady Liberty is 151 ft tall. <br /><br />$151 is the largest prime amount you can make with three distinct US bank notes. <br /><hr /><b> The 152nd Day of the Year</b>: the eighth prime number is 19, and 8 x 19 = 152.... 152 is also the largest known even number that can be expressed as the sum of two primes in exactly four ways. (Students should find all four ways.) <br /><br />152 is a refactorable number since it is divisible by the total number of divisors (8) it has, and in base 10 it is divisible by the sum of its digits(8), making it a Niven number.<br /><br />152 is the sum of four consecutive primes, starting with 31. <br /><br />There are 152 mm tick marks on a six-inch ruler.<br /><br />152 is the smallest number you can make as the sum of two distinct odd primes cubed.<br /><br />the digits 152 occur beginning at the digit of e, that is the 152nd prime number (881). <br /><hr /><b>The 153rd Day of the Year</b>153 The Sum of the aliquot divisors and the product of aliquot divisors are both perfect squares There is a smaller year day with this same property <br /><br />153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits. For more detail and explanation see,<a href="http://pballew.blogspot.com/2012/05/cubic-attractiveness-of-153.html" target="_blank">"The Cubic Attractiveness of 153" </a>,<br /><br />1<sup>3</sup>+5<sup>3</sup> + 3<sup>3</sup> = 153. Numbers which are the sum of their own digits raised to the power of the number of digits are called Armstrong numbers. Except for the trivial one digit numbers, it is also the smallest. There are only three other numbers greater than one which are the sum of the cubes of their digits (Go fourth and seek them. hint: this is the only one which is a year date) <div>(Students who have explored Happy Numbers may enjoy exploring the number chains formed by the sum of the cubes of the digits. Other numbers, like 352 pass through several iterations before closing the loop back to themselves, and of course, some numbers never make it back home. <b>352</b> --> 160 --> 217 --><b>352</b><br /><br />And to extend that, the amazing Cliff Pickover shared this:(although the digits are taken in sets) <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Cf4oL2pbp9w/W5iIby43_HI/AAAAAAAAJK8/bKnA_CARxwQ-CjpWFXdr9ATSrNmtWEbfQCLcBGAs/s1600/153%2Bas%2Bcube%2Bof%2Bdigits.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="440" data-original-width="880" height="160" src="https://3.bp.blogspot.com/-Cf4oL2pbp9w/W5iIby43_HI/AAAAAAAAJK8/bKnA_CARxwQ-CjpWFXdr9ATSrNmtWEbfQCLcBGAs/s320/153%2Bas%2Bcube%2Bof%2Bdigits.jpg" width="320" /></a></div><br /><br />ALSO, 153 = 1! + 2! + 3! +4! +5!, *Jim Wilder@wilderlab </div><div><br /></div><div>Not only is 153 the sum of the cube of its digits, and the sum of five consecutive factorials, it is the sum of the first 17 positive integers, 1 + 2 + .... + 16 + 17 = 153. That makes 153 the 17th triangular number, and its reversal, 351, is also a triangular number, the 26th. <br /><br />I had never observed that 153 = 3 x 51, a product that uses all the digits of the number. HT to INDER J. TANEJA @IJTANEJA There are no other numbers below 1000 that have the same digits as their prime factorization in a simple product (w/o using powers) but there is one lingering just above that number, can you find it? <br /><br />153 also forms a Ruth-Aaron pair with 154, the product of the distinct Prime factors of each sum to the other.<br /><hr /><b>The 154th Day of the Year:</b><br />154 also forms a Ruth-Aaron pair with 153, the product of the distinct Prime factors of each sum to the other.<br /><br />154 is the smallest number which is a palindrome in base 6, [444]<sub>6</sub> ; base 8 ,[242]<sub>8</sub>; and base 9 ,[181]<sub>9</sub> all three. <i>Student's might search for a number that is a palindrome in other simple bases.</i><br /><br />154 also has an interesting property with appropriate powers, 1+5<sup>6</sup>+4<sup>2</sup>= 15642. What other day numbers can you find with similar properties? <br /><br />154 is the twelfth day of the year which is the product of exactly three distinct primes. <br /><br />154 is the number of ways to partition forty into at most, three parts. (It is also the way to partition 43 into parts of which the greatest part is three). <br /><br />If You start with 0! = 1, then 154 is the sum of the first six factorials<br /><br />The largest prime gap below 10,000,000 is 154.<br /><br />154! + 1 is a prime *Prime Curios <br /><br />With just 17 cuts, a pancake can be cut up into 154 pieces. This is called the Lazy Caterers sequence. <br /><br /><hr /><b>The 155th Day of the Year</b><br />The 155th day of the year; 155 is the sum of the primes between its smallest and largest prime factor. 155 = 5 x 31 and (5+ 7 + 11 + 13 + 17 + 19 + 23 + 29 +31 = 155) *Prime Curios <br /><br />Fun with primes: 2^2 + 3! + 5! + 7^2 - 11 - 13 = 155.<br /><br />And from Math Year-Round @MathYearRound 155² +155 ± 1 are twin primes. Students (and teachers) may be surprised how frequently x2+ x ± 1 forms twin primes.<br /><br /> At one time, a new perfect number of 155 digits was announced. On March 27,1936 The Associated Press released a story that a new 155 digit perfect number had been found by Dr. S. I. Krieger of Chicago. The number was \(2^{256}(2^{257} - 1)\) by proving the \(2^{257} -1\) was prime. This was shocking since D. H. Lehmer and M. Kraitcik had announced that the number was composite in 1922. Unfortunately, their method did not include giving a factor of the number. The perfection of the number was doubted by most mathematicians, but the actual factoring to prove it was composite didn't happen until 1952 when the SWAC confirmed it was composite by finding a proper divisor. *Beiler, Recreations in the Theory of Numbers. According to current lists, the closest number of digits for a perfect number are an 77 digit number found by Edouard Lucas in 1876, and a 314 digit number found by R M Robinson in 1952.<br />155 is also a pentagonal number, n*(3*n-1)/2, n=0, +- 1, +- 2, +- 3, ..... Euler showed that the pentagonal numbers are the coefficients of the expansion of the infinite polynomial (1-x)(1-x2)(1-x3).... John H. Conway showed that the same series can be found by taking the triangular numbers that are divisible by three, and dividing them. <br /><br />155 is equal to the sum of the primes from its smallest prime factor, 5, to its largest, 31. There are only three year days of this kind. *HT to Derek Orr<br /><hr /><b>The 156th Day of the Year</b><br />The 156th day of the year; 156 is the number of graphs with six vertices. *What's So Special About This Number. <br /><br />\( ( \pi(1)+\pi(5)+\pi(6)) * (p_1 + p_5 + p_6) = 156 \). 156 is the smallest number for which this is true, and the <i>only</i> even number for which it is true. (The symbols \( \pi(n)\) and \(p_n \) represent the number of primes less than or equal to n, and the nth prime respectively) <br /><br />156 is evenly divisible by 12, the sum of its digits. Numbers which are divisible by the sum of their digits are usually called Niven Numbers. <br /><blockquote>According to an article in the Journal of Recreational Mathematics the origin of the name is as follows. In 1977, Ivan Niven, a famous number theorist presented a talk at a conference in which he mentioned integers which are twice the sum of their digits. Then in an article by Kennedy appearing in 1982, and in honor of Niven, he christened numbers which are divisible by their digital sum “Niven numbers.” One might try to find the smallest strings of consecutive Niven Numbers with more than a single digit. *http://trottermath.net/niven-numbers/ </blockquote>I wonder about the relative order of the classes of numbers which are n times their digit sum for various n.<br /><br />78 is the 12th Triangular number, which means that twice that, 156, is the number of times a clock that chimed the hours would chime in one day. <br /><br />156 is a Harshad (Joy-Giver) number, divisible by the sum of its digits.<br /><br />156/4 = 39 , so 40^2 -38^2 = 156 and 156/12 = 13 so 16^2-10^2 = 156<br /><br />156 is a repunit in base 5 (1111), and a repdigit in base 25 (66) <br /><hr /><b>The 157th Day of the Year </b> <br />2<sup>157</sup> is the smallest "apocalyptic number," i.e., a number of the form 2<sup>n</sup> that contains '666'. *Prime Curios (Can you find an apocalyptic number of the form 3<sup>n</sup>)<br /><br />157 is prime and it's reverse, 751 is also prime. 157 is also the middle value in a sexy triplet (three primes successively differing by six; 151, 157, 163). 751 is also a sexy prime with 757. <br /><br />157 is also the largest solution I know for a prime, p, such that \( \frac{p^p-p!}{p} \) is prime.<br /><br />!57 is a Repunit in Duodecimal, or base 12 (111); and a palindrome in bases 7 (3137) and 12 (11112).<br /><br />The number 157 in base ten is equal to \(31_{[52]}\), but don't worry if you get that backwards, \(52_{[31]}\) is also equal to 157 in decimal. Can you find other examples of reversible numeral/base that give the same decimal value?<br /><br />And from Fermat's Library @fermatslibrary In 1993 Don Zagier found the smallest rational right triangle with area 157. He used sophisticated techniques using elliptic curves paired with a lot of computational power. If he could do that, certainly you ought to be able to find the smallest rational right triangle with area of 1.... (OK trick question, ask your teacher to explain)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-GQQRNFIxAyw/WqIEVmipbQI/AAAAAAAAI5c/wh7LptcXQc4UuNIYUSj4s5MncbQEq6R5ACLcBGAs/s1600/157%2Brational%2Bright%2Btriangle.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="413" data-original-width="673" height="392" src="https://4.bp.blogspot.com/-GQQRNFIxAyw/WqIEVmipbQI/AAAAAAAAI5c/wh7LptcXQc4UuNIYUSj4s5MncbQEq6R5ACLcBGAs/s640/157%2Brational%2Bright%2Btriangle.jpg" width="640" /></a></div><br /><br />157 is the largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1 and, The largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1.*Prime Curios <br /><br />Two to the power 157 is the smallest "apocalyptic number," i.e., a number of the form 2n that contains '666'. *Clifford Pickover<br /><br />157 is the smallest emirP whose sum of the digits (13) is another emirP. And it's the 37th Prime, another emirP<br /><br />157 is the largest prime, p, for which \( \frac{p^p+1}{p+1}\) is prime<br /><br />157 is the smallest three-digit prime that produces five other primes by changing only its first digit: 257, 457, 557, 757, and 857. *Prime Curios<br /><br />28 x 157 = 4396 uses all nine non-zero digits.<br /><hr /><b>The 158th Day of the Year </b><br />The 158th day of the year; 158 is the smallest number such that sum of the number plus its reverse is a non-palindromic prime: 158 + 851 = 1009 and 1009 is a non-palindromic prime. *Number Gossip (What's the next one?)<br /><br />Middle school # 158 in Bayside, Queens, New York, is called Marie Curie Middle School. <br /><br />158 is the sum of the first nine Mersenne prime exponents.<br /><br />The smallest number such that the sum of the number and its reverse is a prime that is not palindromic, i.e., 158 + 851 = 1009.*Prime Curios<br /><br />The decimal expansion of 100! (the product of the first 100 natural numbers) has 158 digits. <br /><br />158 is a number in the Perrin sequence, but lovingly called the "skiponacci" sequence after its resemblance to the Fibonacci sequence. Defined by a(n) = a(n-2) + a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2. The pattern starts 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17,...<br /><hr /><b>The 159th Day of the Year:</b><br />A barrel of oil contains 159 Liters. <br /><br />159 = 3 x 53, and upon concatenating these factors in order we have a peak palindrome, 353, which is itself a prime.*Prime Curios <br /><br />159 is the sum of 3 consecutive prime numbers: 47 + 53 + 59 and can be written as the difference of two squares in two different ways.<br /><br />159 is the fifth Woodall number, a number of the form n*2<sup>n</sup> -1. The numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-defined Cullen numbers n*2<sup>n</sup> +1. )<br /><br />Deshouillers (1973) showed that all integers are the sum of at most 159 prime numbers. I'm waiting for someone to tell me the number that takes 159 prime numbers to form??? </div><div><br /></div><div>48 x 159 = 5346, uses all nine non-zero digits <br /><hr />The 160th Day of the Year <br />160 is the smallest number which is sum of cubes of 3 distinct primes, the first three. (2<sup>3</sup>+3<sup>3</sup>+5<sup>3</sup>) *Prime Curios (It is also the sum of the first power of the first 11 primes )<br /><br />160! - 159! + 158! - ... -3! + 2! - 1! is prime. (Quick, guess the approximate size of this number.)<br /><br />160 is divisible by 4, 8, 16, so \(160 = 41^2 - 39^2 = 22^2 - 18^2 = 14^2 - 6^2 \)<br />And since 160/20 = 8, 160 = = 13^2 - 3^2<br /><br /><br />160 is also the sum of two non-zero squares (12<sup>2</sup> + 4<sup>2</sup>) and like all such numbers, you can show that 160<sup>2n+1</sup> will also be the sum of two non-zero squares.<br /><br />160 is a palindrome in base 3 (12221); and in base 6 (424); <br /><br />160 is the largest year day (and second largest known) for which the alternating factorial sequence is prime: 160!- 159! + 158! - 157! .... + 2! - 1!. <div><span style="background-color: white; color: #222222;">The alternating factorial 5! - 4! + 3! - 2! + 1! = 121. The alternating factorial sequence is prime for n= 3 through 8 (5, 19, 101, 619, 4421, 35899). In spite of this run of consecutive primes, John D Cook checked and found only 15 n values for which the alternating factorial starting with n is prime. 14 are year days, the largest being 160. The one non-year day it turns out uses the same digits as 160, 601. </span></div><div><span style="color: #222222;"><br /></span>In the Collatz problem, starting at 160 takes ten iterations to reach 1, all of them but one is a divided by two step. </div><div><br /></div><div>160 is the sum of the cubes of the first three primes. \(2^3 + 3^3 + 5^3 = 160\)<br /><hr /><b>The 161st Day of the Year:</b><br />Every number greater than 161 is the sum of distinct primes of the form 6<i>n</i> - 1. *Prime Curios (which numbers less than 161 are also the sum of distinct primes of the form 6n-1? or which are not?)<br /><br />and for the gamblers out there, There are <b>161 </b>ways to bet on a roulette wheel.<br /><br />When 161 is not only a palindrome, when is rotated 180<sup>o</sup> it gives a palindromic prime, (191) (Such reversible numbers, or words, are called "ambigrams", among other terms.)<br /><br />Palindrome expression for 161 , 16 + 61 + 7 + 16 + 61<br /><br /><br />161 is the sum of five consecutive prime numbers: 23 + 29 + 31 + 37 + 41 = 161 <br /><hr /><b>The 162nd Day of the Year</b>162 is the smallest number that can be written as the sum of 4 positive squares in 9 ways.*<a href="http://www2.stetson.edu/~efriedma/numbers.html" target="_blank">What's Special About This Number</a>? (Can you find all nine ways?...I should add that five of these use four distinct squares, and the other four have a repeated square..... Can you find a smaller number that can be written as the sum of four squares in eight ways?) [spoiler, the nine ways are shown at the bottom of this entry]<br /><br />the 12th prime (12 = 1*6*2) ; p<sub>12</sub> = 37, and the number of primes less than 162, \( \pi(162)\) is also 37. There is no smaller number with this property. <br /><br />Palindrome expressions for 162 3x3x2x3x3 or 9x2x9<br /><br />162 is the total number of baseball games each team plays during a regular season in Major League Baseball. (But sadly, probably not this year ({2020})<br /><br />Jim Wilder pointed out that 162<sup>1</sup>= 162 has a digit sum of nine; and 162<sup>2</sup>= 26244 has a digit sum of 18; and 162<sup>3</sup>= 4251528 has a digit sum of 27. And 162<sup>4</sup> ??? <br /><br />162 has a sum of divisors 1+2+3+6+9+18+27+54+81=201 which is greater than 162. Such numbers have been called abundant since the Ancient Greeks. <br /><br />A 3x3 magic square with a magic constant for each row and column of 162<br /><br />53 58 51<br />52 54 56<br />57 50 55<br /><br />Imagine you have seven distinctly colored balls, and three numbered tubs to put them in, but none can be in a tub by itself. There are 162 different ways to distribute the balls. (If students struggle with this large a challenge, they can try to find all eleven ways to put five colored balls in just two tubs, again with no solitary balls. <br /><br />A T Vandermonde should be remembered for the wonderfully useful approach he had for generalizations on the factorial, and in my mind created the most useful notation ever (and, he seems to have been the first to think of 0!=1) His notation included a method for skipping numbers, so that [p/3]n would indicate p(p-3)(p-6)... (p-3(n-1)); and in his notation 162 = [9/3]3 or 9*6*3. Now that's a notation worth having an exclamation point. Today this is called a triple factorial, but it doesn't, to my knowledge, have a way to stop along the way, like 16*13*10. <div><br /></div><div><br /></div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-VcIUWNgSqFw/X3SO3ytUu_I/AAAAAAAANME/wOUcvv5YE-M6Nih6J5CE3W6vfq_d6pl6ACLcBGAsYHQ/s900/162%2Bas%2Bfour%2Bsquares.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="371" data-original-width="900" height="251" src="https://1.bp.blogspot.com/-VcIUWNgSqFw/X3SO3ytUu_I/AAAAAAAANME/wOUcvv5YE-M6Nih6J5CE3W6vfq_d6pl6ACLcBGAsYHQ/w610-h251/162%2Bas%2Bfour%2Bsquares.png" width="610" /></a></div><br /> <br /><hr /><b>The 163rd Day of the Year</b><br />The 163rd day of the year; \$ e^{\pi*\sqrt{163}} \$ is an integer. Ok, not quite, In the April 1975 issue of Scientific American, Martin Gardner wrote (jokingly) that Ramanujan's constant (e^(π*sqrt(163))) is an integer. The name "Ramanujan's constant" was actually coined by Simon Plouffe and derives from the above April Fool's joke played by Gardner. The French mathematician Charles Hermite (1822-1901) observed this property of 163 long before Ramanujan's work on these so-called "almost integers." Actually equals <span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">262537412640768743.99999999999925..</span><br /><br /><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><span class="mwe-math-element" face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><img alt="\pi \approx {2^{9} \over 163}\approx 3.1411" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad361b3649c79cf9eb1db987319bf39802ad379" style="border: 0px; display: inline-block; height: 5.676ex; margin: 0px; vertical-align: -1.838ex; width: 18.312ex;" /></span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">. and </span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><span class="mwe-math-element" face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><img alt="e\approx {163 \over 3\cdot 4\cdot 5}\approx 2.7166\dots" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa90c55cce6c1c788511681d86f995547f661798" style="border: 0px; display: inline-block; height: 5.343ex; margin: 0px; vertical-align: -2.005ex; width: 24.532ex;" /> *</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">WIkipedia</span><br /><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><br /></span>Colin Beveridge @icecolbeveridge pointed out that \( (2+\sqrt{3})^{163} \) is also very, very close to an integer. (but it is very large, greater than 10<sup>93</sup> , and was not, to my knowledge, ever the source of an April fools joke.) <br /><br />163 is conjectured to be the largest prime that can be represented uniquely as the sum of three squares \$163 = 1^2 + 9^2 + 9^2 \$.<br /><br />Most students know that the real numbers can be uniquely factored. Some other fields can be uniquely factored as well, for instance, the complex field a+bi where i represents the square root of -1 is such a field. In 1801, Gauss conjectured that there were only nine integers k such that \(a + b\sqrt{-k} \) is a uniquely factorable field.; The largest of these integers is 163.; Today they are called Heegner numbers after a proof by Kurt Heegner in 1952.<br /><br />163 is as easy as 1+2*3^4. <br /><br />163 is a strictly non-palindromic number, since it is not palindromic in any base between base 2 and base 161.<br /><br />163 figures in an approximation of π, in which \( \pi \approx \frac{2^9}{163} \approx 3.1411\). *Wik<br /><br />163 is the 38th prime number<br /><hr /><b>The 164th Day of the Year</b><br /> With the ordered digits of 164 we can form 3 2-digits numbers. Those 3 numbers ± 3 are all prime (16 + 3 = 19, 16 - 3 = 13, 14 + 3 = 17, 14 - 3 = 11, 64 + 3 = 67, 64 - 3 = 61). *Prime Curios<br /><br />In base 10, 164 is the smallest number that can be expressed as a concatenation of two squares in two different ways: as 1 + 64 or 16 + 4<br /><br />There are 164 ways to place 5 non-attacking queens on a 5 by 8 board. */derektionary.webs.com/april-june <br /><br />164 can be expressed as the concatenation of two squares in two different ways, 1, 64 or 16,4. The smallest number for which that is true. Can you think of the next?<br /><br />Because \(\frac{164}{4} = 41, 164=42^2 - 40^2\).<br /><br />164 = 10^2 +8^2<br /><br />A scrabble board has 225 squares on the board, many are special squares with double letter or double word notation, but 164 have nothing.<br /><br />164 is CLXIV in Roman Numerals, using every symbol 100 or below once each.<br /><br />164 is a palindrome in base \(20002_3 \), and in base \(202_9\),</div><div><br /></div><div>T(164) (the 164th triangular number) is the hypotenuse of a triangle with all triangular numbers for its side lengths. The legs of the triangle are T(132) and T (143). \(8778^2 + 10296^2 = 13530^2\)<br /><hr /><b>The 165th Day of the Year</b> <br />The 165th day of the year; 165 is a tetrahedral number, and the sum of the first nine triangular numbers. The tetrahedral numbers are found on the fourth diagonal of Pascal's Arithmetic Triangle, and given by the combinations of (n+2 Choose 3) or Tet<sub>n</sub> = \( \frac {(n)(n+1)(n+2)}{6}\) also easy to remember this is the nth triangular number times (n+2)/3. <br /><br />165 is a sphenic number, the product of three distinct primes.<br /><br />Numbers greater than 35 that end in five are the difference of two squares five apart, 165-25 = 140 so 19^2-14^2 = 165 also, all numbers in the sequence of f(n)= 6n+9 are (n+3)^2 - n^2. so 169 = 209^2 - 26^2<br /><br />165 is also the sum of the squares of the first five odd numbers. <br /><br />165 is the sum of the divisors of the first fourteen integers.<br /><br />165 is sort of a prime average(or an average of primes) The two nearest primes are 163 and 167, with 165 as their average; The next two nearest are 157 and 173, yeah, 165 is their average; The next two out are 179 and 151, yes again, average is 165; then 149 and 181, yep!.... 139 and 191, yep!.... 137 and 193...Oh Yeah!... 131 and 197... awww heck, but if you slipped 199 in in place of 197, you'd get one more. <br /><br />A 5x5 magic square with arithmetic sequence and magic constant of 165<br /><br />41 55 9 23 37<br />53 17 21 35 39<br />15 19 33 47 51<br />27 31 45 49 13<br />29 43 57 11 25<br /><br />165 has three prime factors, called a sphenic number. Its reversal, 561 also has three prime factors. <br /><br />165 is a tetrahedral number, the sum of the triangular numbers from 1 to 45.<br /><br />165 is a palindrome in base 2(10100101) and bases 14 (BB14), 32 (5532) and 54 (3354).<br /><hr /><b>The 166th Day of the Year;</b><br />the reverse (661) of 166 is a prime. If you rotate it 180<sup>o</sup> (991) it is also prime. The same is true if you put zeros between each digit (10606). *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a><br /><br />166!-1 is a factorial minus one prime. (For which n is N! -1 or n! + 1 a prime? <i>hint</i>: there are three more year days for which n! +1 or n! -1 is prime<br /><br />166, like 164, uses all the Roman digits from 100 down, once each. A difference is that 166 uses them in order of their size, CLXVI. <br /><br /><br />166 is a palindrome in base 6 (434) and base 11(141) <br /><hr /><b>The 167th Day of the Year,</b><br />167 is the smallest of a sextet of numbers related to the well known Ramanujan taxicab number. Ramanujan had commented that 1729, the number of Hardy’s taxi, was the smallest number that can be expressed as the sum Of two positive cubes in two ways. But what about three ways? Unfortunately, I don’t know who found this (help?). 167^3 + 436^3 = 228^3 + 423^3= 255^3 + 414^3 =87539319. <br /><br />So here's a challenge, Wieferich proved that 167 is the only prime requiring exactly eight cubes to express it. Can you find the eight? *Prime Curios There are 16 year dates that can not be expressed with 17 non-negative cubes. <br /><br />167^4 = 777796321, the smallest number whose fourth power begins with four identical digits, *Prime Curios <br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-gLxL9ta3E44/XqnAKla_MxI/AAAAAAAAMNw/qP1UAlhM5JsA9a8jBlck5_oziaZVfgVuQCLcBGAsYHQ/s1600/167%2Bmeets%2B169%2BNYC.jpg" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="525" data-original-width="700" height="240" src="https://3.bp.blogspot.com/-gLxL9ta3E44/XqnAKla_MxI/AAAAAAAAMNw/qP1UAlhM5JsA9a8jBlck5_oziaZVfgVuQCLcBGAsYHQ/s320/167%2Bmeets%2B169%2BNYC.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*MAA Found Math</td></tr></tbody></table><br />167 is the 39th prime. <br />It's reversal, 761 is also prime, so 167 is an emirP, and it's the smallest of them whose emirP index, it's the 13th emirp) is also an emirP <br />167 (prime) and 169 (square) meet, at least in New York City <br /><br />Remember, LaGrange said that every natural number can be expressed with four squares. So find them for 167. <br /><br />The reciprocal of 167 is a repeating decimal with a digit cycle of 166 digits. <br /><hr /><b>The 168th Day of the Year </b> <br />there are 168 prime numbers less than 1000. *Prime Curios<br /><br />168 is the product of the first two perfect numbers. *jim wilder @wilderlab<br /><br />\(2^{168} = 374144419156711147060143317175368453031918731001856 \) lacks the digit 2; no larger 2<sup>n</sup> exists for \(n \lt 10^{399}\) that is not pandigital.<br /><br />168/4 = 42 so 168= 43^2-41^2; 168 /8 = 21 so 168 = 23^2 - 19^2, and 168/12 = 14 so 168 = 17^2 - 11^2<br />There are 168 hours in a week.<br /><br />168 is also the number of moves that it takes a dozen frogs to swap places with a dozen toads on a strip of 2(12) + 1=25 squares (or positions, or lily pads) where a move is a single slide or jump. This activity dates back to the 19th century, and the incredible recreational mathematician, Edouard Lucas *OEIS. Prof. Singmasters Chronology of Recreational Mathematics suggests that this was first introduced in the American Agriculturalist in 1867, and I have an image of the puzzle below. The fact that they call it, "Spanish Puzzle" suggests it has an older antecedent. (anyone know more?) <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-99dqsPfJAe8/WwWxDVNSI7I/AAAAAAAAJA8/xD0SlEK7980T86yfrOV8GX96K6Opl0XQgCLcBGAs/s1600/frogs%2Band%2Btoads%2Bpuzzle.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="204" data-original-width="627" height="129" src="https://3.bp.blogspot.com/-99dqsPfJAe8/WwWxDVNSI7I/AAAAAAAAJA8/xD0SlEK7980T86yfrOV8GX96K6Opl0XQgCLcBGAs/s400/frogs%2Band%2Btoads%2Bpuzzle.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">168 and 249 have an interesting relationship, the sum of their digits are equal, and the sum of the squares of their digits are equal. \( 1 + 6 + 8 = 2 + 4 + 9 = 15\) and \(1^2 + 6^2 + 8^2 = 101 = 2^2 + 4^2 + 9^2\) <br /></div><hr /><b>The 169th Day of the Year</b><br /><br />The 169th day of the year; 169 is the smallest square which is prime when rotated 180<sup>o</sup> (691) What is the next one?<br /><br />And from Jim Wilder, 169 is the reverse of 961. The same is true of their square roots... √169=13 and √961=31 or stated another way, 169 = 13<sup>2</sup> and in reverse order 31<sup>2</sup> = 961 <br /><br />An interesting loop sequence within Pi. If you search for 169, it appears at position 40. If you then search for 40, it appears at position 70. Search for 70, ... 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, *<a href="http://www.angio.net/pi/piquery.html" target="_blank">Pi Search page</a><br /><br />169 is the only year day which is both the difference of consecutive cubes, and a square: \(8^3-7^3 =169=13^2\)<br /><br />The first successful dissection of a square into smaller squares was of a square with 169 units on a side. 1907-1914 S. Loyd published The Patch Quilt Puzzle. A square quilt made of 169 square patches of the same size is to be divided into the smallest number of square pieces by cutting along lattice lines. The answer, which is unique, is composed of 11 squares with sides 1,1,2,2,2,3,3,4,6,6,7 within a square of 13. It is neither perfect nor simple. Gardner states that this problem first appeared in 1907 in a puzzle magazine edited by Sam Loyd. David Singmaster lists it as first appearing in 1914 in Cyclopedia by Loyd but credits Loyd with publishing Our Puzzle Magazine in 1907 - 08. This puzzle also appeared in a publication by Henry Dudeney as Mrs Perkins Quilt. Problem 173 in Amusements in Mathematics. 1917 <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Bsr_JoTi1Wk/WLd3n0fb9fI/AAAAAAAAIqc/VL5bMixMYbcKrSL0BilEcER0wIkb-xdjQCLcB/s1600/Mrs%2BPerkin%2527s%2BQuilt.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="313" src="https://4.bp.blogspot.com/-Bsr_JoTi1Wk/WLd3n0fb9fI/AAAAAAAAIqc/VL5bMixMYbcKrSL0BilEcER0wIkb-xdjQCLcB/s320/Mrs%2BPerkin%2527s%2BQuilt.png" width="320" /></a></div><br />169 = (2^7 + 7^2) - (7 + 1) and is the smallest perfect square of the form (2p + p2) - (p + 1).*Prme Curios <br /><br />169 is the sum of seven consecutive primes, 13 + 17 + 19 + 23 + 29 + 31 + 37<br /><br />169 is the last square in the Pell sequence <br /><br />169 is a palindrome in base 12(121) "A gross plus two dozen and one more." <br /><hr /><br /><b>The 170th Day of the Year </b><br />The 170th day of the year; the start of a record-breaking run of consecutive integers (170-176) with an odd number of prime factors.<br /><br />170 is the smallest number that can be written as the sum of the squares of 2 distinct primes, where each of these primes is the square of a prime added to another prime (170 = (2<sup>2</sup> + 3)<sup>2</sup> + (3<sup>2</sup> + 2)<sup>2</sup>). *Prime Curios<br /><br />170 is the largest integer for which its factorial can be stored in double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10<sup>306</sup>. (For 171! it returns "infinity".)<br /><br />170 is the smallest number n for which phi(n)(the number of integers relatively prime to 170=64=8<sup>2</sup>) and sigma(n) (the sum of the divisors of 170=324=18<sup>2</sup>) are both square.<br /><br />Just as a curiosity, the \( \sigma_0(170)\), sometimes called the divisor function is 8, that is, there are eight numbers that divided evenly into 170, 1, 2, 5, 10, 17, 34, 85, and 170, and the number of totitives (coprime values) is 8^2. Some people also use \( \sigma_2(n)\) for the sum of the squares of the divisors of n. For 17, that's 37,700 , and yes, they do similar things for any power that amuses them.<br /><hr /><b>The 171st Day of the Year</b><br /><br /><br />\( 10^{171 } - 171 \)is a prime number with 168 nines followed by 829<br /><br />Google calculator gives 171! = infinity. (close enough in many cases)<br />171 - 9 = 162, and 162/6 = 27, so 30^2 - 27^2 = 171<br /><br />Das Ambigramm added that 163 = 3 x 7 x 11 x 11 x 7 x 3, but also 5 x 5 + 11 x 11 + 5 x 5<br /><br />171 is the only multidigit year-day that is both a triangular number and a palindrome; and it is one of only eleven triangular year days that is divisible by the sum of its digits.<br /><br />It was Gauss who discovered that all natural numbers are the sum of at most three triangular numbers, a discovery he announced with pride on July 10, 1796 when he wrote in his diary, EYPHKA (Eureka), num = \( \bigtriangleup + \bigtriangleup = \bigtriangleup\)<br />171 is the 18th triangular number, the sum of the integers from 1 to 18. <br /><br />171 is a repdigit in base 7 (333)<br /><br />171 is a Harshad (joy-giver) number, divisible by the sum of it's digits.<br /><hr /><b>The 172nd Day of the Year </b><br />seventeen 2's followed by two 17's is prime.*Prime Curios <b>222222222222222221717 is prime</b><br /><br />\( 172 = \pi(1+7+2) * p_{(1*7*2)} \). It is the only known number (up to 10^8) with this property. pi(n) is the number of primes less than or equal to n, and p<sub>n</sub> is the nth prime. <br /><br />172/4 = 43, so 44^2 - 42^2 = 172<br /><br />172 is the sum of Euler's Totient function (the number of smaller numbers for each n, which are coprime to n) over the first 23 integers <br /><br />172 is the number of pieces a circle can be divided into with 18 straight cuts. It is sometimes called the Lazy Caterer's sequence, and is given by the relation \(p = \frac{n^2+n+2}{2}\) <br />Since I haven't mentioned this anywhere else yet, these numbers appear in Floyd's Triangle, a programing exercise for beginning programmers which has the Lazy Caterer sequence going veritcally down the altitude of a triangle of numbers, and the triangular numbers on the hypotenuse <br />1<br />2, 3<br />4, 5, 6<br />7, 8, 9, 10<br />11..... <br /><br />172 is a repdigit in base 6(444), and also in base 42 (44) <br /><hr /><b>The 173rd Day of the year</b> <br />the only prime whose sum of cubed digits equals its reversal: 1<sup>3</sup> + 7<sup>3</sup> + 3<sup>3</sup> = 371. *Prime Curioos <br /><br /> 137 is the sum of the squares of the first seven digits of pi, \(3^2+ 1^2 + 4^2 + 1^2 + 5^2 + 9^2 + 2^2 = 137.\) *Prime Curios (There is no smaller number of digits of pi for which this is true.) If you add the square of the next digit (6^2) you get another prime which is a permutation of the digits of this one, 173. These two are the only prime year days which are the sum of the squares of the first n digits of Pi.</div><div><br /></div><div>Another permutation of 173 is 371, and 173 is the hexdecimal expression of 371 in decimal. </div><div><br />The smallest prime inconsummate number, i.e., no number is 173 times the sum of its digits. (The term inconsummate number was created by John Conway from the Latin for unfinished. [when?])<br /><br />173 is the largest known prime whose square (29929) and cube (5177717) consist of totally different digits. <br /><br />173 is a Sophie Germani Prime since 2*173+1 = 347 is also prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.<br /><br />173 is the sum of the squares of two Fibonacci numbers. (Which two?) and the difference of two squares, 173 = 87^2 - 86^2<br /><br />173 = 1 + 2^2 + 2^3 + 2^5 + 2^7. The exponents are consecutive primes.*Prime Curios<br /><br />And hey, 173 is the prime first three digits of the square root of three (also prime, but you knew that). <br /><br />173 is the sum of three consecutive primes, 53 + 59 + 61 = 173 (Wondering what percent of the time three consecutive primes add up to a prime? Seems fairly common with low digits.) <br />173 is a palindrome in base three(20102) and base 9 (212) <br /><hr /><b>The 174th Day of the Year</b> <br />there are 174 twin prime pairs among the first 1000 integers. <br /><br />174 = 7<sup>2</sup> + 5<sup>3</sup> (using only the first four primes) and is also the sum of four consecutive squares. <br /><br />174 is the smallest number that begins a string of four numbers so that none of them is a palindrome in any base, b, \( 2 \leq b \leq 10 \)<br /><br />174 is a sphenic (wedge) number, the product of three distinct prime factors, 174 = 2*3*29. <br />174 is called an "integer perfect number" because its divisors can be partitioned into two sets with equal sums. <br /><br />174 is the smallest number that can be written as the sum of four distinct squares in six different ways, <br />174 = 1^2 + 2^2 + 5^2 + 12^2<br />= 1^2 + 3^2 + 8^2 + 10^2<br />= 1^2 + 4^2 + 6^2 + 11^2 <br />= 2^2 + 5^2 + 8^2 + 9^2<br />= 3^2 + 4^2 + 7^2 + 10^2<br />= 5^2 + 6^2 + 7^2 + 8^2<br />*Srinivasa Raghava K <hr /><b>The 175th day of the year; </b><br />175 is the smallest number n greater than 1 such that n^6 \(\pm 6\) are both prime. *Prime Curios & Derek Orr <br /><br />175 - 25 = 150 and 150 /10 = 15 so 20^2-15^2 = 175<br /><br />175 is the number of partitions of 35 into prime parts. <br /><br />From Jim Wilder @wilderlab : \( 175 = 1^1 + 7^2 + 5^3 \) There is one more three digit year date which has this same relation. Find it.<br /><br />A normal Magic square of order 7 has a "magic constant" of 175 for the sum of each row, column or diagonal. The one below comes from the "Geeks For Geeks" web site, but this particular Geek wishes they had rotated it one-quarter turn clockwise so that the smallest number is in the center of the bottom row.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-RRVA9wLluLY/XrxBqQtfERI/AAAAAAAAMZk/tTlWEw-hC0kDVOucmC3Wi1VqOimI20inwCLcBGAsYHQ/s1600/Magic%2Bsquare%2B7x7.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="237" data-original-width="411" height="184" src="https://1.bp.blogspot.com/-RRVA9wLluLY/XrxBqQtfERI/AAAAAAAAMZk/tTlWEw-hC0kDVOucmC3Wi1VqOimI20inwCLcBGAsYHQ/s320/Magic%2Bsquare%2B7x7.jpg" width="320" /></a></div>And if you want a unique way to create any normal magic square (and with a little imagination, lots of other odd order magic squares) for a nice way to create the one above , but rotated <a href="https://pballew.blogspot.com/2018/06/suprise-unique-approach-for-odd-order.html">https://pballew.blogspot.com/2018/06/suprise-unique-approach-for-odd-order.html</a><br /><br /><br /><hr /><b>The 176th Day of the Year</b><br />176 and its reversal 671 are both divisible by 11. ( Students should confirm that the reverse of any number that is divisible by 11 will also be divisible by 11.)<br /><br />176 is a happy number, repeatedly iterating the sum of the squares of the digits will lead to 1, 12 + 72 + 62= 86, 82 + 62 = 100 and 12 + 02 + 02 = 1 <br /><br />The number 15 can be partitioned in 176 ways. For younger students, imagine all the different ways of making fifteen cents with US coins, 1 cent, 5 cnets, and 10 cents.... now imagine there were also coins worth 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, and 15 cents. There would be 176 different collections of coins that would total exactly 15 cents. <br /><br />The divisors of 176 are 1, 2, 4, 8, 11, 16, 22, 44, 88, and 176. Take away 1, 176, 8 (the sum of the four 2's that are prime factors, and 11, (the other prime factor) now add up what's left, 2+4+16+22+44+88= 176, yeah, making magic happen here. <br /><br />176 is a Self number, it can't be written by any other number plus the sum of its digits. 21 for instance, is not a self number because 15+1+5 = 21. <br /><br />8*20 + 16 = 176 so 176 = 24^2 - 20 ^2 (try this on 192 also ) and it is divisible by 16 with quotient 11, so 15^2 - 7^2 = 176.<br /><br />176 is also a happy number, if you square each of its digits, and add them, then do the same to the result, eventually you will have the happy ending of getting 1, which would simply repeat itself forever, and knowing this, you will stop... please stop.... STOP!<br /><br />176 is also a cake number, the number of ways of slicing a cube with 10 planes to get the greatest number of cake pieces. A three dimensional analogy to the lazy caterer's number in two space. <br /><hr /><b><span style="font-family: inherit;">The 177th Day of the Year</span></b><br /><span style="font-family: inherit;">there are 177 </span>graphs with seven edges. *What's So Special About This Number. (<i>only 79 of these are connected graphs)</i><br /><br /><ul><li>177 is the smallest magic constant for a 3 x 3 prime magic square</li></ul><img alt="\begin{bmatrix} 17 & 89 & 71 \\ 113 & 59 & 5 \\ 47 & 29 & 101 \end{bmatrix}." class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/3/e/6/3e661cfddf061d801390421d5d3b4949.png" /><br /><br /><span style="font-family: inherit;">177 is the sum of the primes from 2 to 47 taking every other one, 2+5+11+.... +41+47, and 177 is the 15th or 1+7+7th prime. *Prime Curios</span><br /><br />177 is a deficient number, the sum of its aliquot divisors is 59+3+1 = 63, far less than 177. Its deficiancy is 114= 177-63. All odd numbers up to 945 are deficient, the second smallest deficient odd number is 1575 All of the known abundant odds are divisible by 5.<br /><br />177 is also a Leyland number, expressible as a^b + b^a. both greater than one . using 2 and 7 in this case. . The numbers are named for British mathematician Paul Layland from Oxford University. There are only nine year days that are Leyland numbers. Only one of those nine is prime.<br /><hr /><b>The 178th Day of the Year</b><br />178 = 2 x 89. Note that 2 and 89 are the smallest and the largest Mersenne prime exponents under 100. *Prime Curios<br /><br />178 is a palindrome in base 6,\( [454]_6 \) and in base 8 \([262]_8\)<br /><br />Strangely enough, 178 and 196 are related... In fact, 178 has a square with the same digits as 196: 178<sup>2</sup> = 31,684 196<sup>2</sup> = 38,416 178 has also a cube with the same digits as 196: 178<sup>3</sup> = 5,639,752 196<sup>3</sup> = 7,529,536 *Zoo of Numbers<br /><br />178 = 13^2 + 3^2<br /><br />178 is a palindrome in base 6 (454), base 7 (343), and base 8 (262) <br /><br />178 is a semi-prime, the product of 2 and 89, which are the smallest, and largest Mersenned prime exponents under 100. <br /><br />178 is a digitally balanced number, it binary expression has an equal number of zeros and ones, 10110010, and they are balanced so that the first and last, 2nd and 2nd last, etc always have a one and a zero. <br /><hr />The 1179th Day of the Year<br />179 is a prime whose square, 32041, has one each of the digits from 0 to 4. <br /><br />179 is a <a href="http://pballew.blogspot.com/2013/05/knockout-primes-and-new-notation.html" target="_blank">"Knockout Prime"</a> of the form K(3,2) since 17, 19, and 79 are all prime.<br /><br />179 is an emirp, a prime whose reversal, 971 is also prime, and the combination sum and product 179 * 971 + 179 + 971=174959 is also an emirp. <br /><br />179<sup>3</sup> has all odd digits, 5735339. *Derek Orr<br /><br />179 = (17 * 9) + (17 + 9)<br /><br />A winning solution to the 15-hole triangular peg solitaire game is: (4,1), (6,4), (15,6), (3,10), (13,6), (11,13), (14,12), (12,5), (10,3), (7,2), (1,4), (4,6), (6,1). The term (x,y) means move the peg in hole x to y. Not only does this solution leave the final peg in the original empty hole, but the sum of the peg holes in the solution is prime. But not just any prime, it is 179. <br /><br />Between the beginning and the 179th digit of π, an equal number of five different decimal digits occur (there are 18 each of the digits 0, 3, 4, 5, and 9). Mike Keith conjectures this to be the last digit of π for which this happens (there are no others up to 10^9 digits). *Prime Curios <br /><br />1/179 has a repeating patter of 178 digits, called a full repetend prikme. <br /><br />179 is a strictly non-palindromic number. It is not a palindromic number in any base.*Wikipedia <br /><hr /><b>The 180th Day of the Year</b><br />180 can be formed with the only the first two primes... 180 = 2<sup>2</sup> x 3<sup>2</sup> x (2+3) *Prime Curios<br /><br />180 is the sum of two square numbers: \( 12^2 + 6^2 \). It can also be expressed as either the sum of six consecutive primes: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive primes: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. <br />As differences of two squares 180= 46^2 - 44^2<br /><br />180 is a Harshad (Joy-Giver in Sanskrit) as it is divisible by the sum of its digits.<br /><br />Beautiful trigonometry, arctan1 + arctan2 + arctan3 = 180<sup>o</sup><br /><br />180=2^2*3^2 *(2+3)<br /><br />180 has more divisors than any smaller number. It is also called a refactorable number because it is divisible by the number of divisors it has, 18. <br /><br />Pi radians is equivalent to 180<sup>o</sup><br /><br />As was 178, 180 is a digitally balanced number, its eight binary digits contain four ones and four zeros, 10110100, and like 178, they match up into two sets of balanced zero-one pairs, the first four digits, 1011, aligning perfectly with their digital opposite in the last four 0100. <br /><hr /></div></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-69513020852574765372020-04-16T15:28:00.030-07:002021-05-24T12:29:52.633-07:00Number Facts for Every Year Date (121-150) <b>The 121st Day of the year:</b><br />121 is the smallest square that requires five powers of two for a sum.\( 2^6 + 2^5 + 2^4 + 2^3 + 2^0 \) No smaller square needs more than three. The next square year day requiring five will be in the last week of December.<br /><br /><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">121 will be the largest year day of the form n!+1 which is a square number. Brocard conjectured in 1904 that the only solutions of n! + 1 = m</span><sup style="background-color: white; color: #222222;">2</sup><span style="background-color: white; color: #222222;"> are n = 4, 5, and 7. There are no other solutions with </span> </span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br />The alternating factorial 5! - 4! + 3! - 2! + 1! = 121. The alternating factorial sequence is prime for n= 3 through 8 (5, 19, 101, 619, 4421, 35899). In spite of this run of consecutive primes, John D Cook checked and found only 15 n values for which the alternating factorial starting with n is prime. 14 are year days, the largest being 160. The one non-year day it turns out uses the same digits as 160, 601. </span></span></span><div><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span></span></div><div><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span></span><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">121 is also the only square of the form 1 + p + p</span><sup style="background-color: white; color: #222222;">2</sup><span style="background-color: white; color: #222222;">+ p</span><sup style="background-color: white; color: #222222;">3</sup><span style="background-color: white; color: #222222;"> +p</span><sup style="background-color: white; color: #222222;">4</sup><span style="background-color: white; color: #222222;">. where p is prime. Find the value of n. </span></span>Other such squares, if they exist, must exceed 35 digits.<span style="font-family: inherit;"><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">121 is a Smith Number, a composite number for which the sum of its digits is equal to the sum of the digits in its prime factorization. Smith numbers were named by Albert Wilansky of Lehigh University. He noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith: </span></span></span><span style="background-color: white; color: #222222; font-family: inherit;">4937775 = 3 × 5 × 5 × 65837, while 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42.</span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">There are 49 Smith numbers below 1000, collect the whole set.</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">121 is a palindrome in base ten, and also in base 3 (11111), base 7 (232) and base 8(171). No other year day is a base ten palindrome and also palindrome in as many other (2-9) bases.</span></span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">121 is a palindromic number that is the square of another palindromic number. Several others should be easy to find.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Fermat conjectured that 4 and 121 are the only numbers of the form n^3-4. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">You can write 121 as the sum of a prime and its reversal in three different ways. Can you find them?</span><br /><span style="font-family: inherit;"><br /></span></span><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-_5STTjojHm8/XnqKH01SaaI/AAAAAAAALoo/-C5Eyfh0fIIGhe8M69QOklm-ayidOCPawCLcBGAsYHQ/s1600/Chinese_checkers%2Bboard.png" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="220" data-original-width="220" src="https://1.bp.blogspot.com/-_5STTjojHm8/XnqKH01SaaI/AAAAAAAALoo/-C5Eyfh0fIIGhe8M69QOklm-ayidOCPawCLcBGAsYHQ/s1600/Chinese_checkers%2Bboard.png" /></a></td></tr><tr><td class="tr-caption" style="font-size: 12.8px; text-align: center;">*Wikipedia</td></tr></tbody></table><span style="font-family: inherit;"><span style="font-family: inherit;">A <b>star</b> number, is a number for the set of points that would be in the interior of a </span>Chinese<span style="font-family: inherit;"> checker table in which the "home" triangles are of size n. The star number for the standard board with ten in each home triangle has 121 = 5+6+7+8+9+8 +7+6+5 points. (Chinese checkers are neither Chinese, or Checkers, but fun anway.) </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Every number greater than 121 is the sum of distinct primes of the form 4n+1. </span><br /><span style="font-family: inherit;"><br /></span>121 is the smallest composite palindrome for which a permutation of the digits, 211, is prime. *Prime Curios<br /><br />121 in base 3 is a repdigit (11111), and a palindrome in base ten and base 8 (171), and is 3D in base 36.<br /><br /><span style="font-family: inherit;">On a personal point, M-121 was originally the name of the major east-west highway across Michigan's Upper Peninsula, now US 2. Passing about four miles south of my wife's retreat home near Rexton.</span></span><br /><hr /><span style="font-family: inherit;"><br /><b>The 122nd day of the year</b><br /> there are 122 different ways to partition the number 24 into distinct parts. Euler showed that this is the same as the number of ways to partition a number into odd parts. One distinct way would be (12, 6, 3, 2, 1) , five distinct numbers, and one odd way would be (3,3,3,3,3,3,3,3) with eight odd parts, or (21,3) with only two odd parts.<br /><br />122 ends in the digit two when written in base 3, 4, 5, 6, 8, 10, 12, 15, and 20. How unusual is that?</span></div><div><span style="font-family: inherit;"><br /></span></div> 122 squared minus each of its prime factors squared is also prime <br /> and 122 is the smallest sum of two non-consecutive factorials of distinct primes (2! + 5!) *Prime Curios<br /><br /><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;">Not sure how unusual this is, but there are no twin primes between 121^2 and 122^2? </span></span><div><span style="color: #212529;"> </span></div><div><div><hr /><span style="font-family: inherit;"><b>The 123rd Day of the Year:</b><br />The number formed by the concatenation of odd numbers from 123 down to 1 is prime. (ie 123121119...531 is Prime) *Prime Curios (Who figures stuff like this out???) </span><br /><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;"> Japan Airlines Flight 123, was the world's deadliest single-aircraft accident in history. </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;"> And here is an interesting curiosity from the archimedes-lab.org/numbers file: Write down any number (excluding the digit 0): 64861287124425928 Now, count up the number of even and odd digits, and the total number of digits it contains, as follows: 12 | 5 | 17 </span></div><div><span style="font-family: inherit;"> Then, string those 3 numbers together to make a new number, and perform the same operation on that: 12517 1 | 4 | 5 </span></div><div><span style="font-family: inherit;"> Keep iterating: 145 1 | 2 | 3 You will always arrive at 123.</span></div><div><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">123 is the tenth Lucas Number, named for Eduoard Lucas who studied and extended the similar Fibonacci numbers, and was the creator of the fascinating Towers of Hanoi puzzle. </span><br /><span style="font-family: inherit;"><img alt="{\displaystyle L_{n}=\varphi ^{n}+(1-\varphi )^{n}=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n}\,,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb560884b301a295339a5674eb4f4f2ec81295ab" style="background-color: white; border: 0px; color: #222222; display: inline-block; font-family: sans-serif; font-size: 14px; height: 6.509ex; vertical-align: -2.505ex; width: 68.468ex;" /></span><br />*Srinivasa Raghava K@SrinivasasR1729<br /><br /></div><div><span style="font-family: inherit;">There are only two positive integers that are both two more than a perfect square, and two less than a cube, 123 = 11^2 + 2 and 5^3 - 2. The other should be easy to find. </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">And some of us remember a popular IBM software spreadsheet from the '80s, Lotus 123. </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">And 123 is the difference of two squares in two different ways, 62² - 61² and 22² - 19². The pattern of both these are explained in Day 111. <br /></span><br /><hr /><span style="font-family: inherit;"><br /><b>The 124th Day of the Year</b><br />124 =σ( 1! * 2! * 4!) *Prime Curios (The <b>sigma function</b> of a positive integer <i>n</i> is the sum of the positive divisors of <i>n)</i> </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 124 is also an Odious number: a number with an odd number of 1's in its binary expansion.(just recently, it occurred to me that it would be more appropriate if an Odious number, had an odd number of "0's") </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 124 in base two is expressed as 1111100. (Easy to see 31*4 in that)</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> ±1 ± 2 ± 3 ± 4 ± 5 ± 6 ± 7 ± 8 ± 9 ± 10 ± 11 ± 12 = 0 has 124 solutions (collect the whole set) *Math Year-Round @MathYearRound</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">124 is the sum of eight consecutive primes..... find them children.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">In all the infinity of positive integers, there is not one of them whose proper divisors add up to 124; such numbers are called untouchable. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">124 in base five is a repdigit, 444, which means it's one less than 5^3 (students new to studying bases should observe that a repdigit k units long in base n, will always be (n^k)-1.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">And in case it's not obvious, the digits of 124 form an exponential sequence, called the doubling sequence, 1, 2, 4.. </span></div><div><br /></div><div>124 is divisible by four so it is the difference of two squares of numbers that differ by 2, and since 124 / 4 = 31, the numbers must straddle 31, 32² - 30² = 124. <br /><hr /><span style="font-family: inherit;"><b>The 125th Day of the Year</b><br /><br />125 is a cube, and the sum of distinct squares (and these are distinct PRIMES squared.) There is no smaller value for which this is true. 125 = 5<sup>3</sup> = 11<sup>2</sup> + 2<sup>2</sup> What's the next? It can also be 10^2 + 5^2 .</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">125 can also be written as a curious sort of palindrome, 125 = 5<sup>(2+1)</sup> *Jim Wilder, @wilderlab</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> Conjectured by Zhi-Wei Sun to be the largest power (5<sup>3</sup>) for which there is no prime between it and the previous power (11<sup>2</sup>). </span><span style="font-family: inherit;">The other prime gaps between powers are in (2</span><sup style="font-family: inherit;">3</sup><span style="font-family: inherit;">, 3</span><sup style="font-family: inherit;">2</sup><span style="font-family: inherit;">), (2</span><sup style="font-family: inherit;">5</sup><span style="font-family: inherit;">, 6</span><sup style="font-family: inherit;">2</sup><span style="font-family: inherit;">) and (5</span><sup style="font-family: inherit;">2</sup><span style="font-family: inherit;">, 3</span><sup style="font-family: inherit;">3</sup><span style="font-family: inherit;">). </span><br /><br /><span style="font-family: inherit;">125 and 126 are a Ruth Aaron pair of the second kind. In the first kind </span>prime<span style="font-family: inherit;"> factors are only counted once, in the second kind they are counted as often as they appear, so 5+5+5 = 2+3+3+7. Some Ruth-Aaron pairs only have one of each factor, so they qualify under either method. The original kind were discovered for 714 Ruth's career record, and 715, the number on the day Aaron passed his record (he went on to get more). </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">125 is a palindrome in base 4(1331) and in base 20, with the cool name of the vigesimal (from the Latin 'vicesimus', the French 'vingt' is still used for naming some number between 70 and 99) system (65). 20 is also a score, so if someone asks the day of the year, it's six score and five. </span><br /><hr /><span style="font-family: inherit;"></span></div><div><span style="font-family: inherit;"><b>The 126th Day of the Year</b><br />nine points around a circle form the vertices of \( \binom{9}{4} = 126 \) unique quadrilaterals. That also means that if you draw all the diagonals of the nonagon, you would be using the same 126 sets of four vertices to get 126 intersections.<br /><br /><span style="font-family: inherit;">as 126 = 125 + 1 it is almost obvious that it is the sum of two cubes. It is also the sum of a cube and a square, and it is the first of four consecutive numbers that are the sum of a cube and a square. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> In non-leap years, there are 126 days in which the day of the month is prime.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> The prime gap that covers the first century with no primes (from 1671800 to 1671899) has length 126 (from 1671781 to 1671907). </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">The sum of the unique prime factors of 126 is equal to the product of its digits, there is no smaller multi-digit number for which this is true. *Prime Curios</span><br /><br /><span style="font-family: inherit;"> There are 9 choose 5, or 126 ways for a random selection to pick the five spaces on a tic tac toe board for the "first player" in a random game. 36 of these configurations are a "win" for both players. They have both three x's and three O's in a line, since they don't have an order of play. Over 58% of those games are a win for the "first player". Geometrically, a student could think of each random game as a pentagon selected from nine points spaced around a circle. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">126 is a palindrome in base 5 (1001) and in base 20(66), six score and six.<br /><br />The "magic numbers" in Physics are the numbers of nucleons (protons are neutrons) which exactly fill the shells, and thus form very stable isotopes. The numbers are 2, 8, 20, 28, 50 82 and <b>126</b>. The first six numbers correspond to the elements, helium, oxygen, calcium, nickel, tin, lead, and the element for 126 has not, at this writing, been discovered.<br /><span style="font-family: inherit;"><span style="font-family: inherit;"><br />125 and 126 are a Ruth Aaron pair of the second kind. In the first kind </span>prime<span style="font-family: inherit;"> factors are only counted once, in the second kind they are counted as often as they appear, so 5+5+5 = 2+3+3+7, The original kind were discovered for 714 Ruth's career record, and 715, the number on the day Aaron passed his record (he went on to get more). </span> </span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><br /></span></span><span style="font-family: inherit;"><span style="font-family: inherit;">On the 126th day of the year 1937, the Hindenburg Zeppelin crashed in New Jersey, (See Day 129)</span><br /><hr /><span style="font-family: inherit;"><b>The 127th Day of the Year</b></span></span><br /><span style="font-family: inherit;">127 is the last </span>prime <span style="font-family: inherit;">year day that will be a repdigit in base 2 (1111111)</span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><br /></span></span><span style="font-family: inherit;"><span style="font-family: inherit;">126 was the sum of two cubes, 127 is the concatenation of two cubes, 1, 27. </span></span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><br /></span></span><span style="font-family: inherit;"><span style="font-family: inherit;">The fourth perfect number, 8128, is 127 * 64 , which relates to :</span></span><br /><span style="font-family: inherit;">127 is the fourth Mersienne Prime, 2</span><sup style="font-family: inherit;">7</sup><span style="font-family: inherit;">-1. Édouard Lucas verified 2</span><sup style="font-family: inherit;">127</sup><span style="font-family: inherit;">-1 as prime in 1876, and it remained the largest known prime for over 70 years. He is said to have spent 19 years in checking this 39 digit prime by hand. This remains the largest prime number discovered without the aid of a computer. (Lucas also invented the </span><a href="http://pballew.blogspot.com/2009/08/tower-of-hanoi.html" style="font-family: inherit;" target="_blank">Towers of Hanoi </a><span style="font-family: inherit;">Puzzle, and the game of dots and boxes which he called "La Pipopipette".) </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 2</span><sup style="font-family: inherit;">0</sup><span style="font-family: inherit;"> + 2</span><sup style="font-family: inherit;">1</sup><span style="font-family: inherit;"> + 2</span><sup style="font-family: inherit;">2</sup><span style="font-family: inherit;"> + 2</span><sup style="font-family: inherit;">3</sup><span style="font-family: inherit;"> + 2</span><sup style="font-family: inherit;">4</sup><span style="font-family: inherit;"> + 2</span><sup style="font-family: inherit;">5</sup><span style="font-family: inherit;"> + 2</span><sup style="font-family: inherit;">6</sup><span style="font-family: inherit;"> = 127.</span><br /><span style="font-family: inherit;"><br /></span></span></span>French Mathematician Alphones de Polignac is know for two conjectures about prime numbers; the first was that any odd number greater than two could be formed by sum of a power of two and a prime. He was wrong. His statement is now known to be false, as 127 can not be so formed. Although false, his conjecture may be true for all composite numbers. Every exception I have found, like 127 and 149 and 251, are primes. </div><div>His other conjecture is related to the idea of twin primes. He conjectured that for every even number, there are an infinite number of primes that distance apart. So far this one is still unproven. </div><div><br /></div><div><br /></div><div><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/--yagD9xESro/Xn50ewVi_aI/AAAAAAAALqk/ulHQf-e_PPwQslhzd0uVNOunVgMh6jeOwCLcBGAsYHQ/s1600/road%2Brunner.jpg" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="126" data-original-width="167" src="https://1.bp.blogspot.com/--yagD9xESro/Xn50ewVi_aI/AAAAAAAALqk/ulHQf-e_PPwQslhzd0uVNOunVgMh6jeOwCLcBGAsYHQ/s1600/road%2Brunner.jpg" /></a></td></tr><tr><td class="tr-caption" style="font-size: 12.8px; text-align: center;">*Cartoonspot.net</td></tr></tbody></table><span style="font-family: inherit;">Wile E Coyote's arch nemesis, the Road Runner, is one of 127 species of the Cukkoo.<br /><br /><span style="font-family: inherit;">127 is the sum of the first nine odd primes, 3+ 5+ 7 +.... + 29 = 127</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">The last three digits of the 11th Mersenne Prime, \( 2^{89}-1 = 162259276829213363391578010288127\) .... ends in 127 , the Mersenne Prime, ends in 127. The next Mersenne Prime is M(127). *Prime Curios. </span><br /><span style="font-family: inherit;"><br /></span>127 is a palindrome in base 2 of course, all ones, but also in base 9 (121)<br /><br />127 primes fall between 2000 and 3000. *Prime Curios<br />127 = 1! + 3! + 5!<br /><br />If you find all the prime pairs that add up to 1000, there are 127 of them. *Prime Curios<br /><br />127 mm = 5 inches. Handy reference.<br /><br /><span style="font-family: inherit;">127 can be expressed as the sum of factorials of the first three odd numbers (1! + 3! + 5!). And in a rare equivalence, 127 cm is equal to 50 inches. HT Don S. McDonald @McDONewt</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">127 is the smallest odd prime that can't be written as a prime P + 2^x for some integer. </span><br /><span style="font-family: inherit;"><br /></span>The Prime Gap of 14 days between 109 and 127 is the Longest Prime Gap in the year Days. There will be three such gaps this year, but the next is about four months away, so lots of prime days in your near future.</span><br /><hr /><span style="font-family: inherit;"></span></div><div><span style="font-family: inherit;"><b>The 128th Day of the Year</b> </span><br /><span style="font-family: inherit;">128 is The largest known even number that can be expressed as the sum of two primes in exactly three ways. (Find them) *Prime Curios How many smaller numbers (and which) are there that can be so expressed? </span><br /><span style="font-family: inherit;">But, it can not be expressed as the sum of distinct squares, for any number of squares.<br />And it is the largest such number, ever.... no, I mean EVER. The very last. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">128 = 2^8, so in binary it is a 1 followed by 7 zeros, which makes it also 4^4, and in base 4 its a 2 with three zeros. But it's also 8^2, so in base eight its a 2 with two zeros, </span></div><div>128 is a power of two, and all of its digits are powers of two. I don't know of any other.</div><div><br /><span style="font-family: inherit;"> 128 is also the largest number that cannot be expressed as the sum of distinct squares. *Number Gossip. (Surprisingly, there are only 31 numbers that can not be expressed as the sum of distinct squares. )</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 128 can be expressed by a combination of its digits with mathematical operators thus 128 = 2<sup>8 - 1</sup>, making it a Friedman number in base 10 (Friedman numbers are named after Erich Friedman, as of 2013 an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida.) </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 128 the sum of the factorials of the first three prime numbers, 2! + 3! + 5! =128.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Some nice relationships between 128 and its digits, 128 + (1+2+8) = 139, a prime number. But 128 + (8 + 1) is 137, also prime, and 128 + (2 + 1) is 131, a prime, AND 128 +( 8+2 ) is not prime, but 138 is between a twin prime pair. ..... And 1*2*8 = 16 is a divisor of 128. </span><br /><span style="font-family: inherit;">And that pair of cousin primes, 127 and 131, are the largest such pair with a power of two (128) between them. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">The name for a particular 7th dimensional Hyperplex with 128 vertices is a Hepteract. Dazzle your friends. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Oh, I told you 128 is the 7th power of two.... but there are no more three digit numbers that are 7th powers... </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">And if you like to keep score, 128 is 6 score and 8. In old commercial terminology, a schock was a lot of 60 items, so 128 is also two shock and 8, or 28 in sexigesimal (base sixty). The number of stalks of corrn or wheat (supposedly) gathered and stood on ends in the fields to dry, like in "When the frost is on the Pumpkin and the Fodders in the shock. " </span></div><div><br /></div><div><br /></div><div>128 is divisible by four so it is the difference of two squares of numbers that differ by 2, and since 128 / 4 = 32, the numbers must straddle 31, 33² - 31² = 128. </div><div>But it is also divisible by eight, so it is the difference of two squares of numbers that differ by four(there is a power of two relation working here, which students might find). And since 128/8 = 16, 18² - 14² = 128<br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">And in 1968 the 128 K Mac was the hottest desktop computer around. </span><br /><hr /><span style="font-family: inherit;"><b>The 129th Day of the Year</b><br />129 is the smallest number with four representations as a sum of three positive (but not necessarily distinct) squares: 129 = 1<sup>2 </sup>+ 8<sup>2 </sup>+ 8<sup>2 </sup>= 2<sup>2 </sup>+ 2<sup>2 </sup>+ 11<sup>2 </sup>= 2<sup>2 </sup>+ 5<sup>2 </sup>+ 10<sup>2 </sup>= 4<sup>2 </sup>+ 7<sup>2 </sup>+ 8<sup>2 </sup>. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 129 is also the sum of the first ten primes. </span><br /><span style="font-family: inherit;"><br />129 is the fourth number in a row that is the sum of a square and a cube, and 129 that may be accomplished in two different ways. </span></div><div><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 129 is the smallest sum of distinct seventh powers (1<sup>7</sup> + 2<sup>7</sup>).</span><br /><span style="font-family: inherit;">It's also \( 2^7 + 2^0 \) and thus a palindrome in base two, (10000001) and a repdigit palindrome in base 6(333).</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">If you concatenate the prime divisors of 129, 3 and 43, you get 343, you get a prime to a prime power, \( 7^3\)</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">129 is a Happy Number, if you sum the square of the digits, and continue iterating that process on each new number, you end up at one. Unhappily, the origin of the name and the creator are unknown. It was popularized by Leeds Math Professor Reg Allenby, who heard about it from his daughter who picked it up at school. Suspected origin is Russia according to the late Richard Guy. </span><br /><br /><span face="sans-serif" style="color: #222222;"><span style="background-color: white; font-family: inherit;">And if you've not spent some time in Western Ky, and perhaps even if you have, you might not guess where the official Banana Capital of the US is. It's in the little town of Fulton, Ky, along the train route from New Orleans to Chicago, and Fulton had the distinction of being the place where Union Fruit company chose to pause the trains bringing fresh bananas along the way to re-ice them for the rest of their journey.At one time over 70% of Bananas shipped into the US came through Fulton. About 13 miles away is the even smaller town of Wingo, formerly called Wingo Station ( because it set along the same New Orleans and Ohio rail line passing through Fulton. And what they have in common other than that, is the reason I mention them today, they are on the ends of Ky Route 129. They are just a pretty spring drive of 40 miles from here in Possum Trot. </span></span><br /><span face="sans-serif" style="color: #222222;"><span style="background-color: white;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">"<span style="font-size: large;">Oh, the humanity, and all the passengers screaming around here!"</span></span></span><span face="sans-serif" style="background-color: white; color: #222222;"> Herbert Morrison broadcasting live over WLS Chicago from NAS Lakehurst New Jersey as he reported on the burning of the Hydrogen filled Zeppelin, The Hindenburg, on May 6, 1937 (that was the 126th day of that year), but it's number was LZ129.. </span></span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-pDS8r8nY0hA/Xn-QOjsZD6I/AAAAAAAALq0/8PbGUl3vB4oUIb0ghDgVwp1Z4hgOGkpdQCLcBGAsYHQ/s1600/Hindenburg_disaster%252C_1937.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="261" data-original-width="220" src="https://1.bp.blogspot.com/-pDS8r8nY0hA/Xn-QOjsZD6I/AAAAAAAALq0/8PbGUl3vB4oUIb0ghDgVwp1Z4hgOGkpdQCLcBGAsYHQ/s1600/Hindenburg_disaster%252C_1937.jpg" /></a></td></tr><tr><td class="tr-caption" style="font-size: 12.8px;">*Wikipedia</td></tr></tbody></table><br /><hr /><span style="font-family: inherit;"><b>The 130th Day of the Year</b><br />130 is the sum of the factorials of the first five terms of the Fibonacci sequence. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 130 is the sum of the squares of its four smallest divisors, ( \( 1^2 + 2^2 + 5^2 + 10^2 = 130 \) </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">130 is also the <b>only</b> number equal to the sum of the squares of its first 4 divisors: 130 = 1^2 + 2^2 + 5^2 + 10^2.*Prime Curios </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> This is the 46th day of the year that is the sum of two squares, 3^2 + 11^2. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">130 is the sum of consecutive odd powers of 5, 5^1 + 5^3</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Haven't mentioned the hexgonal numbers much this year so far, but 130 is the largest number that cannot be written as the sum of four hexagonal numbers.</span><br /><span style="font-family: inherit;">Hexagonal numbers are given by the formula H(n) = n(2n-1), and produce the sequence 1, 6, 15, 28, 45, 66, 91... (can you find numbers that ARE the sum of four ) (All the even perfect numbers are in that sequence..) </span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-rcNT5RNnQls/XoFfFm2cnRI/AAAAAAAALr4/eFJ6F4mvXUwfcsrMlROS5saX0S7hll0HgCLcBGAsYHQ/s1600/hexagonal%2Bnumbers.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="125" data-original-width="368" height="107" src="https://1.bp.blogspot.com/-rcNT5RNnQls/XoFfFm2cnRI/AAAAAAAALr4/eFJ6F4mvXUwfcsrMlROS5saX0S7hll0HgCLcBGAsYHQ/s320/hexagonal%2Bnumbers.png" width="320" /></a></td></tr><tr><td class="tr-caption" style="font-size: 12.8px;">*Wikipedia</td></tr></tbody></table><span style="font-family: inherit;">130 is a palindrome in base 3 (11211), and in base 4(2002), and in base 8(202), and in base 12 (AA) (which is 10 twelves + 10)</span><br /><hr /><br /><span style="font-family: inherit;"><b>The 131st Day of the Year</b><br />131 is the sum of three two-digit primes (31 + 41 + 59) whose concatenation is the decimal expansion of pi (3.14159...). </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> Any ordering of the digits of 131 is still prime. This is called an "absolute" prime and a permutable prime. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> 131 is the sum of three prime numbers that all begin with the same digit. *Prime Curios </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> bonus: 131 is the 32nd prime and the sum of the digits of both numbers is 5. </span><br /><span style="font-family: inherit;">32 & 131 is the smallest n, P(n) pair with this property. Such numbers are often called Honaker Primes after G. L. Honaker, Jr, . There is only one more such prime that is a year day. </span></div><div>Since 2 (131)+ 1 = 263 is also prime, 131 is called a Sophie Gerrmain prime. Sometimes the prime created by the 2p+1 process is also a Sophie Germain Prime, for example 11 is SGP since 2x11+1 = 23 is prime, and 23 is a SGP since 223+1 = 47 is also prime. <br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> The reciprocal of 131 repeats with a period of 130 digits, 1/131 =0.007633587786259 54198473282442748091603053435114503816793893129770992366412213740458 015267175572519083969465648854961832...</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">131 is the smallest integer for which </span><span style="background-color: white; color: #212529; font-size: 16px;">the sum of its digits in every </span><a class="glossary" href="https://primes.utm.edu/glossary/xpage/Radix.html" style="background-color: white; border: 1px dashed rgba(0, 51, 0, 0.25); box-sizing: border-box; color: #003300; cursor: pointer; font-size: 16px; padding: 0px 2px; transition: all 0.2s ease-in-out 0s;" title="glossary">base</a><span style="background-color: white; color: #212529; font-size: 16px;"> B = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 is prime . \(10000011_2, 11212_3, 2003_4\) and the digit sums are 3, 7, and 5... all primes. </span></span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span></span><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"> </span></span> <span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">The 131st Fibonacci</span><span style="background-color: white; color: #212529; font-size: 16px;"> (1066340417491710595814572169) is the smallest Fibonacci </span><a class="glossary" href="https://primes.utm.edu/glossary/xpage/FibonacciPrime.html" style="background-color: white; border: 1px dashed rgba(0, 51, 0, 0.25); box-sizing: border-box; color: #003300; cursor: pointer; font-size: 16px; padding: 0px 2px; transition: all 0.2s ease-in-out 0s;" title="glossary">prime</a><span style="background-color: white; color: #212529; font-size: 16px;"> which contains all the digits from 0 to 9. *Prime Curios</span></span></span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">US 131 is a road mostly through Michigan ending at the beautiful town of Petoskey, on the shores of Lake Michigan. Important as the birthplace of Claude Shannon, the father of information theory. </span></span></span><br /><span style="font-family: inherit;"><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">Think like Fibonacci, 131 = 0! + 1! + 1! + 2! + 3! +5!</span></span></span></div><div><br /></div><div>One fact about 131 reminds me of some early number theorems, Find a number that divided by four has a remainder of three, and divided by 3 has a remainder of 2, and divided by 2 has a remainder of 1.... </div><div><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span><span style="background-color: white; color: #212529; font-size: 16px;">131 is a knockout prime, since you can "knockout' any digit and leave a prime, 31, 11, 13. Chris Maslanka created the term. on a twitter feed with me. The symbol for a number like 131 where the three digits form a prime number is KP(3,3), a three digit prime; and with any one crossed out, they still form a prime, so it's KP(3,2) and if all three of the digits were prime, it would be KP(3,1) but we no longer accept one as prime. I used KP for knockout Prime over another good suggestion. It is also a Permutable prime for which PP(3,3) is an appropriate symbol. For the story of my "discovery" see <a href="https://pballew.blogspot.com/2013/05/knockout-primes-and-new-notation.html">Knockout Primes, and a New Notation. </a></span></span><br /><hr /></div><b>The 132nd Day of the Year</b><br /><br />132 and its reversal (231) are both divisible by the prime 11 (132/11 = 12, 231/11 = 21). Note that the resulting quotients are also reversals. *Prime Curios<br /><br /> 132 is the last year day which will be a Catalan Number. The Catalan sequence was described in the 18th century by Leonhard Euler, who was interested in the number of different ways of<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-qRHH0rWMC2U/XoNlohIYrwI/AAAAAAAALwo/NjkmklwM2nY-_g4uAi_t1D-rXG9ohTnWACLcBGAsYHQ/s1600/tohanoi.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="320" data-original-width="306" height="320" src="https://1.bp.blogspot.com/-qRHH0rWMC2U/XoNlohIYrwI/AAAAAAAALwo/NjkmklwM2nY-_g4uAi_t1D-rXG9ohTnWACLcBGAsYHQ/s320/tohanoi.jpg" width="305" /></a></div>dividing a polygon into triangles (the octagon can be divided into 6 triangles 142 ways. The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle.<br /><br />132 is the smallest number with this property,<a href="http://en.wikipedia.org/wiki/132_%28number%29#cite_note-1"></a><br /><br />132 and its reversal (231) are both divisible by the prime 11 (132/11 = 12, 231/11 = 21). Note that the resulting quotients are also reversals. *Prime Curios 132 is the last year day which will be a Catalan Number. The Catalan sequence was described in the 18th century by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles (the octagon can be divided into 6 triangles 142 ways. The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle.<br /><br /> If you take the sum of all 2-digit numbers you can make from 132, you get 132: <img alt="12 + 13 + 21 + 23 + 31 + 32 = 132" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/9/b/1/9b1c5e3f6c2f62cb1c07b882c0c9c87b.png" />. 132 is the smallest number with this property. The other two are both multiples of 132. </div><div>Students, it should be obvious that all permutations of 1,2,3 will produce the same result. But can you see what 015, 033, 141, 222 will also go to the same absorbing state. Use a similar analysis to find numbers that end in 264.<br /><br />132 = 2 * 3 * 11, these three factors can be arranged in three orders to produce a prime, 2311, 2113, and 1123. (and of course, no arrangement of the original three digits can form a prime ) and of all the 12 permutations of the digits of the three factors, there are 7; (1123, 1213, 1321, 2113, 2131, 2311, and 3121) that are all prime.<br />And speaking of the factors 11, 2, 3, a nice palindromic expression for 132 is 11*2*3+3*2*11<br /><br />132 is a Harshad (Joy-Giver) number, since it is divisible by the sum of its digits.</div><div>It is also called a refactorable number because it is divisible by the number of its divisors, 12.<br /><br />132 is also a self number, as there is no number n which added to the sum of the digits of n is equal to 132.<br /><br />132 is not a palindrome in any base 2-12, but in base 7(246) it has digits that are each the double of the digits in 132. (I just noticed that, and wonder how often something like that happens?)<br /><hr /><b>The 133rd Day of the Year</b><br />133 is a "happy number". If you sum the squares of the digits and then repeat the process and the sum will eventually come to one. (1<sup>2</sup> + 3<sup>2</sup>+3<sup>2</sup>= 19 ... === 82 === 68 === 100 ====1) Some numbers, "unhappy ones", never reach one. (Student's might explore happy numbers to find how many times the process must be iterated for different numbers to reach one, for example <i>I</i> (33) = 5 Alternatively, curious students may wonder what happens to the "unhappy" numbers if they never reach one.)<br /><br />133 is also a Harshad (Joy-Giver) number, since it is divisible by the sum of its digits.<br /><br />133 is a repdigit in base 11 (111) and base 18 (77),<br /><br /> 133 is the sum of the squares of the first three semi-primes, and is a semi-prime itself. it is the smallest number with this property.<br /><br />According to one classification, there are 133 species of mammal on the Earth, and 1/7th of them are Bats. *Number Freak, Derrick Niederman<br /><br /> 133= 4<sup>2</sup> + 6<sup>2</sup> +9<sup>2</sup><br /><br /> And Jim Wilder @wilderlab posted this interesting observation about 133 and it's reversal, 331.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-FybzgdBn27E/Wvg-0zj8kvI/AAAAAAAAI_M/yE8SA-OWiy8auryXtaK5cK0gXpAetwATwCLcBGAs/s1600/331%2B133%2Bprime%2Bcomposite.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="434" data-original-width="652" height="213" src="https://2.bp.blogspot.com/-FybzgdBn27E/Wvg-0zj8kvI/AAAAAAAAI_M/yE8SA-OWiy8auryXtaK5cK0gXpAetwATwCLcBGAs/s320/331%2B133%2Bprime%2Bcomposite.jpg" width="320" /></a></div><br /><br />133, and 134 were used by Euler in generating birectangular Heronian tetrahedra. He created a method for deriving them from equal sums of fourth powers \$ p^4 + q^4 = r^4 + s^4\$ and used 133 and 134 on one side, and 59 and 158 on the other. The actual side lengths of the three perpendicular edges created from this quartet were over 332,000,000.<br /><br />133 is the smallest integer, n, for which 10 n +(1or 3 or 7 or 9) are all composite. Prime Curios<br /><br />The Dewey Decimal system classification for numerology is 133.533, and if you add the first to the reverse of the second 133+335=666.... <br /><hr /><b>The 134th Day of the Year</b><br />133, and 134 were used by Euler in generating birectangular Heronian tetrahedra. He created a method for deriving them from equal sums of fourth powers \( p^4 + q^4 = r^4 + s^4\) and used 133 and 134 on one side, and 59 and 158 on the other. The actual side lengths of the three perpendicular edges created from this quartet were over 332,000,000.<br /><br />134 has only two prime factors (67 and 2){called a bi-prime or a semiprime, it is the 45th semiprime of the year to date.} . Note that 134<sup>2</sup> - 67<sup>2</sup> = 13467, which is the base numbers concatenated. *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a><br /><br />134 is the sum of <sub>8</sub>C<sub>1</sub> + <sub>8</sub>C<sub>3</sub> + <sub>8</sub>C<sub>4</sub><br /><br />134 is the 19th day of the year that is the sum of three positive cubes.<br /><br />And 134 is t he maximal number of regions the plane can be divided with 12 circles.<br />It is not possible to append a single digit to 134 and produce a prime. ><br /><br />In this politically charged atmosphere, individuals in the military might want to be aware that, the American <b>UCMJ</b>;is the catch-all article, for offences "not specifically mentioned in this chapter." Used to prosecute a wide variety of offences, from cohabitation by personnel not married to each other to statements critical of the U.S. President. Some prisoners, including Abu Ghraib were tagged with this number. Wik<br /><hr /><b>The 135th Day of the year. </b><br /><span style="font-family: inherit;">135 is the smallest non-trivial SP (sum times product) number. If you take the sum of the digits of a number, and also the product of the digits, and then multiply the two outcomes, there are only three positive numbers for which you will get the original value. One works, trivially. The other two are 135 and 144. 135-> (1+3+5)*(1*3*5) = 9*15=135. 144->(1+4+4)*(1*4*4)= 9 * 16 = 144.</span><br /><span style="font-family: inherit;">A Good exercise for students is to take the SP product in a iteration to find out if it goes to zero, or repeats some pattern, or lands eventually on one of these three fixed points. (Try it with your students). 23->5*6 = 30. 30-> 3*0 = 0.... fixed point.</span></div><div><br /></div><div>\(135 \equiv 3 (\mod 6)\) and so 135 is expressible as the difference of two squares, using bases three apart. The two bases must sum to 135 / 3 = 45. So 21 and 24 should work, and 24² - 21² = 135 <span style="font-size: 13px;"> </span><br /><br />135 = 3^3 * 5, with only two distinct factrors, 3 and 5. If you start with three, and square 5 consecutive integers, \(3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 135\) *Prime Curios<br /><br />135 = 1<sup>1</sup> + 3<sup>2</sup> + 5<sup>3</sup>. *<a href="http://www2.stetson.edu/~efriedma/numbers.html" target="_blank">What's So Special About This Number </a> (can you find others?) There are only two year dates that have this property. The second is larger and may take you forty days and nights to find.<br /><br />135 is also equal to (1 x 3 x 5)( 1 + 3 +5) (can you find another number which is either the product or sum of the two factors formed by the product of its digits, and the sum of its digits?) Both 1!+3!+5!= 127 is prime; and 1!!+3!!+5!! is prime if the double factorial n!! means n (n-2)(n-4).... (the same symbol is sometimes used for the factorial of n! ).<br /><br />135 is the "partition number", or the number of ways to partition 14. We still do not know if there are an infinite number of "partition numbers" which are divisible by 3, although we do know there are an infinite number divisible by 2<br /><br />\( 135 = 11 x^2 + 11x +3\) is a simple quadratic that was an important element in Apery's proof that <span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> </span><span class="mwe-math-element" face="sans-serif" style="background-color: white; color: #222222;"><span style="border-color: initial; border-image: initial; border-style: initial; font-size: 14px; height: 2.843ex; vertical-align: -0.838ex; width: 4.067ex;"><img alt="\zeta (3)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3088978098c7b90b2754a9d9b0b994d873e1755c" style="border: 0px; display: inline-block; height: 2.843ex; margin: 0px; vertical-align: -0.838ex; width: 4.067ex;" /></span> </span><span class="mwe-math-element" style="background-color: white; color: #222222; font-family: inherit;">is irrational. To four decimal digits accuracy, it is 1.2020.... the year I'm writing this. </span><br /><br />Quick! How many primes from 1000 to 2000? 135, of course!<br /></div><div><br />The interior angles of a regular Octagon are \(135^o\)<br /><br />This day in Roman numerals is not suitable for minors, CXXXV.<br /><br />135 is a palindrome in base 6( 343) , and base 7(252)<br /><br />Two planets that are \( 135^o \) apart are called sesquiquadrate, and it is said they are in astrological aspect. The aspect terms seems to have been created by Johannes Kepler. And for students, the other big long word means one and one half quadrants. I'll give you another of those sesquipedalian words down the way a bit.<br /><br />135 = 1^1 + 3^2 + 5^3. The only other examples I know of are 175, 518, and 598. </div><div><br /></div><div>The function \( (11 x 10^k + 19)/3 \) generates some unusual primes. If you plug in 135, you get a number beginning with two followed by 135-2 sixes and ending in 73. For example with k=4 you get 36673. Not sure what leads to functions like this, but I have come across several of them over the years. </div><div><br />And if you need to get out of Indianapolis and run down to the Blue Grass, Indiana State Road 135 runs to the boarder on the Ohio at Mauckport, just west of Louisville. Pretty (mostly) two lane roads with lots of little towns and pretty farms.<br /><hr style="font-weight: bold;" /><span style="font-family: inherit;"><b>The 136th Day of the Year</b><br />136 is "power friendly" with 244. \(244 =1 ^3 + 3^3 + 6^3\) and \(136 = 2^3 + 4^3 + 4^3\)</span></div><div><span style="font-family: inherit;">(Only one other number pair share this relation. Can you find them?) </span><br /><span style="font-family: inherit;"> </span></div><div><span style="font-family: inherit;">The sum of all prime factors of 136 is equal to the reversal of \$ \pi(136)\$. \$ \pi(n)\$ is the number of primes less than n<i> (so \(\pi(136)=32 \) </i>and the sum of the prime factors of 136<i> is 2+2+2+17 =23, the revrsal of 32. </i></span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">136 is the number of walks of length 9 between two adjacent vertices in the cycle graph C_8 (A,B,C,D,E,F,G,H)</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">136 is a factor of \( N_Edd\) , the Eddington number, calculated by Arthur Eddington in 1938 for the number of protons in the observable universe. The number can be factored into 136 (2^156) which is about 2^80. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Lots of numbers are expressible as the sum of two squares, but 136 is the smallest that can be expressed where neither of the two are prime. *Prime Curios</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">136 is a triangular number, which means it's the sum of consecutive integers from 1 to n, Find n. (hint: get rid of the prime digit.) </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">In binary 136 is written as the concatenation of two binary "Eights". (10001000). In base 9 is a palindrome (161) and in Hexdecimal it is a repdigit (88). </span><br /><hr /><b style="font-family: inherit;">The 137th Day of the Year</b><br /><span style="font-family: inherit;">137 is the sum of the squares of the first seven digits of pi, 3<sup>2</sup>+ 1<sup>2</sup> + 4<sup>2</sup> + 1<sup>2</sup> + 5<sup>2</sup> + 9<sup>2</sup> + 2<sup>2</sup> = 137. *Prime Curios (There is no smaller number of digits of pi for which this is true.) If you add the square of the next digit (6^2) you get another prime which is a permutation of the digits of this one, 173.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">137 is the third term in a sequence of primes that can be created by staring with 7 and creating a new term by adding a single digit to the front of the previous term; 7, 37, 137 ... It is possible to create a sequence of 15 Prime numbers in this way. <a href="http://oeis.org/A012885?utm_medium=referral&utm_source=t.co" target="_blank">OEIS</a> </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"> Like palindromes, Don McDonald reminded me that 10/137 is a nice one, the period eight repeating palindrome .07299270...</span><br /><span style="font-family: inherit;">If you just use 1/137 you get<span style="font-family: inherit;"> </span></span><span style="background-color: white; color: #212529; font-size: 16px;"><span style="font-family: inherit;">0.00729927.... which </span></span><span style="font-family: inherit;">was also thought to be the fine structure constant in physics according to Eddington.It turned out he was very close, but not quite exact. What if you tried 100/137?</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Wolfgang Pauli died in hospital room 137, after a lifetime trying to prove that 137 was the fine structure constant. It's close, but not so. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">137 is the largest prime factor </span><span style="background-color: white; color: #212529; font-size: 16px;">of 123456787654321*Prime Curios</span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">137 is the 33rd Prime number and is a twin prime with 139, it's a Pythagorean prime, 11^2 + 4^2, and it is a KnockoutPrime (3,2) it remains prime if you knock out any one character leaving two. </span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">137 is not a palindrome in any base between 2 and 135.... Called a strictly non-palindrome. </span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white;"><span style="color: #212529; font-family: inherit;">137 is the first of twelve consecutive primes with equal gaps around the center, sort of a </span><span style="color: #212529;">palindrome</span><span style="color: #212529; font-family: inherit;"> of gaps to make up for being a non-palindrome. The 11 gaps between them is 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, ending in 191. </span></span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">137 divides 11111111, and all the other eight digit repdigits. </span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #212529; font-size: 16px;">And direct from Prime Curios, and coffee loving mathematicians everywhere, </span></span><span style="background-color: white; color: #212529; font-size: 16px;"><span style="font-family: inherit;">The full chemical name for caffeine is 1,3,7-<b>trimethylxanthine.</b></span></span><br /><hr /><span style="font-family: inherit;"><b>The 138th Day of the Year:</b><br />138 is a sphenic number(the product of three primes from the Greek for "wedge shaped") and is the smallest product of 3 primes, such that in base 10, the third prime is a concatenation of the other two: </span><span style="font-family: inherit;">(2)(3)(23)</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">138 is the sum of four consecutive primes (29 + 31 + 37 + 41), </span><br /><span style="font-family: inherit;">and 138 can be written in palindromic expression, 138 = 19+2*7*2+91 *@AmbrigrammDesign. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">138 is an Ulam Number, a member of the sequence created by a sieve process by Stan Ulam in 1964. It begins with the numbers 1, 2, and then each successive term is the smallest larger number that is the sum of two distinct numbers in the sequence, in a single way. The first few numbers are 1, 2, 3, 4, 6, 8, 11,,, Five is missing because its sum can be created in two different ways, 2+3 or 1+4.</span><br /><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">138 is a palindrome in base 8(212)</span><br /><hr /><span style="font-family: inherit;"><b>The 139th Day of the Year:</b></span></div><div><span style="font-family: inherit;"><b><br /></b>The 139th day of the year; 139 and 149 are the first consecutive primes differing by 10. *David Wells, Curious and Interesting Numbers. </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">139 = 9*8+7*6+5*4+3*2-1 *<a href="http://primes.utm.edu/curios/page.php?short=139" target="_blank">Prime Curios </a></span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">139 is the sum of five consecutive prime numbers( 19+ 23+ 29 +31+ 37) </span></div><div><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">139 = 3x3+11x11+3x3 *@AmbigrammDesign</span><br /><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;"> 139 is also a Happy number, A happy number is a number deﬁned by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay),or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).For example, 19 is happy, as the associated sequence is:<br />1^2 + 9^ 2 = 82<br />8^ 2 + 2^ 2 = 68<br />6 ^2 + 8^2 = 100 1^ 2 + 0^2 + 0^2 = 1<br />The happy numbers up to 1,000 are: 1, 7, 10, 13, 19, 23, 28, 31, 32,44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167,176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291,293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368,376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490,496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649,653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761,763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888,899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970,973, 989, 998, 1000.<br /><br />The happiness of a number is unaffected by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number. I propose the use of the term principle Happy numbers for those that do not contain a zero, or an reordering of a previous happy number. That would reduce the above list to the following 31, and makes searches more direct since no descending sequences of digits can exist. 1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899, 998,(The earliest I have ever found this term was in an article in the Bulletin of the California Mathematics Council in 1970. Does anyone know of an earlier usage?")</span></div><div><br /></div><div>139 is the smallest factor of the 23rd Lucas number, It is the smallest Lucas number with prime index which is not prime itself. </div><div><br /></div><div>139 is the sum of five consecutive primes, 19, 23, 29, 31, and 37.139 is the smallest factor of the smallest Lucas number to prime index, which is not prime. </div><div><br /></div><div><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">139 and 149 are the first consecutive </span><a class="glossary" href="https://primes.utm.edu/glossary/xpage/Prime.html" style="background-color: white; border: 1px dashed rgba(0, 51, 0, 0.25); box-sizing: border-box; color: #003300; cursor: pointer; font-family: roboto, sans-serif; font-size: 16px; padding: 0px 2px; transition: all 0.2s ease-in-out 0s;" title="glossary">primes</a><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"> differing by 10 *Curious and Interesting Number by David Wells.</span></div><div><span face=""roboto" , sans-serif" style="color: #212529;"><br /></span></div><div><span face=""roboto" , sans-serif" style="color: #212529;">139 is the smallest prime for which the product of the digits is a prime cubed. </span></div><div><span face=""roboto" , sans-serif" style="color: #212529;"><br /></span></div><div><span face=""roboto" , sans-serif" style="color: #212529;">139 is the larger of a pair of twin primes.</span></div><div><span face=""roboto" , sans-serif" style="color: #212529;"><br /></span></div><div><span face=""roboto" , sans-serif" style="color: #212529;">If you take all the possible square pyramidal numbers, and subtract one, it seems that only three of these can ever be prime, 14-1, 30-1, and 140-1.</span></div><div><span face=""roboto" , sans-serif" style="color: #212529;"><br /></span></div><div>139 is the smallest prime that is the sum of six distinct squares. 1^2 + 2^2 + 3^2 + 5^2 + 10^2. </div><div></div><div><span style="font-family: inherit;"><br /></span><br /><hr /><span style="font-family: inherit;"><b>The 140th Day of the Year;</b><br />140 is the sum of the squares of the first seven positive integers. 1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> + 7<sup>2</sup> = 140. *Prime Curios (and 7 is the largest prime factor of 140 And the 7 consecutive numbers starting with 140, all have an even number of prime factors, 140 is 2x2x5x7. </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">As the sum of the first seven consecutive squares, it is the 7th square pyramidal number. .</span></div><div>And how about an palindromic expression for 140, 2x5x7+7*5*2. from *@AmbigrammDesign<br /><br /></div><div><span style="font-family: inherit;">140 is a repdigit in bases 13 (aa), 19(7,7), 27(5,5), 34(4,4), 69(2,2), and 139(1,1). </span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;">140 is the fourth Harmonic divisor Number, the harmonic mean of its divisors. The harmonic mean is 12 divided by the sum of the reciprocals of the divisors (the 12 is because there are 12 of these divisors) \( \frac {12}{1/2 + 1/2 + 1/4 ...... 1/140} \) =5 All perfect numbers are Harmonic divisor numbers but the converse is not true. All Harmonic divisor numbers are Practical numbers, since some distinct subset of its proper divisors can be used to sum to any smaller number.</span></div><div><span style="font-family: inherit;"><br /></span></div><div><span style="font-family: inherit;"> There are 140 x 10<sup>21</sup> (140 followed by 21 zeroes) different configurations of the Rubik's Cube. *Cliff Pickover@pickover </span></div><div><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">140 is the magic constant of this 5x5 square by *Srinivasa Raghave K Can you see how to easily make one for day 135, or 145? Jeff Miller's Web site on the Earliest Use of Math Words says that Frenicle de Bessy used the term magic in the title to his book, Des quarrez ou tables magiques, published posthumously in 1693, twenty years after his death. The first use in English was the same year in "A New Historical Relation of the Kingdom of Siam." Appropriate to have de Bessey mentioned here, as he first noted the cubic relation of the Taxi-cab number, 1729 and Srinivasa is a big fan, I believe, of Ramanujan.</span><br /><div class="separator" style="clear: both; text-align: center;"><span style="font-family: inherit;"><a href="https://1.bp.blogspot.com/-DykIBg2N9i4/XsQmqWtBqsI/AAAAAAAAMd4/4G5KrwvYSPM52fnt_ZuVjwn_LQKPRtecgCLcBGAsYHQ/s1600/Screenshot_2020-05-19-13-29-10_kindlephoto-862679510.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="603" data-original-width="900" height="214" src="https://1.bp.blogspot.com/-DykIBg2N9i4/XsQmqWtBqsI/AAAAAAAAMd4/4G5KrwvYSPM52fnt_ZuVjwn_LQKPRtecgCLcBGAsYHQ/s320/Screenshot_2020-05-19-13-29-10_kindlephoto-862679510.png" width="320" /></a></span></div><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"><br /></span></div><div> 140 is the character limit on Twitter (or was)<br /> A. J. Meyl proved in 1878 that only three tetrahedral numbers are also perfect squares,The largest of these is T(48) =140<sup>2</sup> = 19600: T(1) = 1² = 1; T(2) = 2² = 4<br /><div><br /></div><div>140 is also a Harshad, or Joy-giver number, divisible by the sum of its digits.<br /><div><br />A semi-magic knights tour is a knights tour in which the 64 numbers on the board are numbered, and the numbers the knight lands on in order are filled in along the rows of an 8x8 magic square grid. A semi-magic square is one where the rows and columns add up to the magic constant, but the diagonals do not. The first such square was produced by William Beverley in 1848. The first known true magic square was not achieved until 2003 by J C Meyriandgnac and G. Sternterbrink, who also showed that there are 140 different Semi-magic tours. *Number Freak, Derriek Nieberman.<br /><hr /><b> The 141st Day of the Year:</b><br />The 141st day of the year; 141 is the first non-trivial palindrome appearing in the decimal expansion of Pi, appearing immediately after the decimal point, 3.14159. Tanya Khovanova, Number Gossip<br />141 is a palindrome in base ten, and also in base six (353)<br /><br />141 is the second n to give a prime Cullen number (of the form n*2<sup>n</sup> + 1). Cullen numbers were first studied by Fr. James Cullen in 1905. (That prime is 393050634124102232869567034555427371542904833,) *David Wells, Curious and Interesting Numbers. 141 is the only Prime Cullen index below 1000. (strangly if you change the +1 to -1, you get lots of index numbers that produce primes, 2, 3, 6, 30... LOTS)<br /><br />141 is the number of lattice paths from (0,0) to (6,6) using steps (2,0), (0,2), (1,1).<br /><br />141 is the 31st Lucky Number. Lucky Numbers were introduced to the public in 1956 by Gardner, Lazurus, Metropolis and Ulam. They suggested naming the sieve that defines it as a Josephus Flavius sieve, because it resembled the counting out sieve in the Josephus problem from the 1st century. The sieve begins by counting out every second number and eliminating them (thus eliminating all the evens). Then counting again from the start, eliminate every nth number where n is the next number in the list after the first survivor. It should proceed something like this:<br />1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21...<br />1, 3, x, 7, 9, x, 13, 15, x, 19, 21, x<br />1, 3, 7, 9, 13, 15 x 21<br />Lucky searching to you all. Seems like a really good computer project for young programming students.<br /><hr /><b>The 142nd Day of the Year:</b><br />there are 142 possible planar graphs with six vertices.<br /><br />142 is the smallest Semi-prime (having exactly 2 prime factors), whose sum of divisors is a cube. 142+71+2+1 = 6<sup>3</sup><br />The binary representation of 142 has the same number of zeros and ones.<br /><br />Using only the digits 1, 4, and 2, and a plus sign:<br />1+42+24+1 is a palindrome, and 1+4+1 is a palindrome, so<br />1 + 42 + 24 + 1 + 1 + 4+ 1 + 1 + 42 +24 + 1 is a palindrome, and equals 142.<br /><br />142 is the number of ways of partitioning 25 into distinct parts... which must be the number of ways of partitioning them into odd parts according to Euler.<br />A pound is 453.59 grams. An ounce is 1/16 of that, or 28.349.. grams. A carat is .2 grams, so an ounce is about, but not quite, 142 carats. Number Freak, Derrick Nieberman<br /><br /></div></div></div><span style="font-family: inherit; font-size: small;">There are 142 planer graphs with unlabeled vertices.<br />Bus 142 (the "Magic Bus"), whose number is clearly visible on the bus that Christopher McCandless lived in until his death in Alaska, features prominently on the bus in the film made about his life called Into the Wild</span><br /><hr /><span style="font-family: inherit; font-size: small;"><br /><b>The 143rd Day of the Year</b><br />there are 143 three-digit primes.<br /><br />Also, 143^2 is a divisor of 143143.HT to Matt McIrvin who found the pattern for numbers such that n^2 divides n.n (where the dot represents concatenation) and then found it is at OEIS I should point out that every number greater than one for which this is true involves the digits 143, in order, and includes a mystery offering from one-seventh.<br /><br />143 is also the number of moves that it takes 11 frogs to swap places with 11 toads on a strip of 2(11) + 1 squares (or positions, or lily pads) where a move is a single slide or jump. This activity dates back to the 19th century, and the incredible recreational mathematician, Edouard Lucas *OEIS.<br />Prof. Singmasters Chronology of Recreational Mathematics suggests that this was first introduced in the American Agriculturalist in 1867, and I have an image of the puzzle below. The fact that they call it, "Spanish Game" suggests it has an older antecedent. (anyone know more?)</span><br /><div class="separator" style="clear: both; text-align: center;"><span style="font-family: inherit; font-size: small;"><span style="font-family: inherit; font-size: small;"><a href="https://1.bp.blogspot.com/-PQ0vvOv-86U/XpdErzmfw2I/AAAAAAAAL64/HahrSQWcLXUQOKxY4NvMBIWhB8pW1G9OQCLcBGAsYHQ/s1600/frogs%2Band%2Btoads%2Bpuzzle.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="204" data-original-width="627" height="208" src="https://1.bp.blogspot.com/-PQ0vvOv-86U/XpdErzmfw2I/AAAAAAAAL64/HahrSQWcLXUQOKxY4NvMBIWhB8pW1G9OQCLcBGAsYHQ/s640/frogs%2Band%2Btoads%2Bpuzzle.jpg" width="640" /></a></span></span></div><span style="font-family: inherit; font-size: small;"><br /><br />143 is the smallest composite number, n, such that 5^n + 2 is prime. (That's some big number) *Prime Curios<br /><br />143 is the sum of seven consecutive primes, beginning with 11.<br /><br />There is no decimal number n, such that n + (sum of its digits) = 143.<br /><br />Waring's problem tells us that every number is the sum of at most 143 seventh powers.<br /><br />\( 3^2 + 4^2 = 5^2 \) AND<br /><br />\( 3^3+4^3+5^3=6^3 \)<br /><br />BUT<br /><br />\(3^4+4^4 + 5^4+6^4=7^4-143\) Another beautiful pattern, spoiled by an ugly truth!<br /><br />143 is a repdigit in base 12 (BB) (that's eleven twelves plus eleven)</span><br /><hr /><span style="font-family: inherit; font-size: small;"><b>The 144th Day of the year.</b><br /><br />144 is the largest possible SP (sum times product) number. If you take the sum of the digits of a number, and also the product of the digits, and then multiply the two outcomes, there are only three positive numbers for which you will get the original value. One works, trivially. The other two are 135 and 144. 135-> (1+3+5)*(1*3*5) = 9*15=135. 144->(1+4+4)*(1*4*4)= 9 * 16 = 144.<br />.A Good exercise for students is to take the SP product in a iteration to find out if it goes to zero, or repeats some pattern, or lands eventually on one of these three fixed points(that's four fixed points if you count zero). (Try it with your students). 23->5*6 = 30. 30-> 3*0 = 0.... fixed point.<br /><br /><br /><br />144 is the only non-trivial square in the Fibonacci Sequence.In fact, there are only four Fibonacci numbers that are perfect powers, 0, 1, 8, and 144. And we haven't known that for so very long. Here is the story from , <a href="http://amzn.to/1U4MdXf" target="_blank">Professor Stewart's Incredible Numbers</a></span><br /><div class="separator" style="clear: both; text-align: center;"><span style="font-family: inherit; font-size: small;"><a href="https://3.bp.blogspot.com/-KpYkSHxBP14/VuhC1jtY9gI/AAAAAAAAIE8/ssxz5FIP0KAKh3sP2jlUJq7_81ihUlOhw/s1600/Fibonacci%2Bpowers%2Bhistory.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="264" src="https://3.bp.blogspot.com/-KpYkSHxBP14/VuhC1jtY9gI/AAAAAAAAIE8/ssxz5FIP0KAKh3sP2jlUJq7_81ihUlOhw/s640/Fibonacci%2Bpowers%2Bhistory.jpg" width="500" /></a></span></div><span style="font-family: inherit; font-size: small;"><br /><br />In 1913 R. D. Carmichael proved his conjecture that for any Fibonacci Number F(n), greater than F(12)=144, has at least one prime factor that is not a factor of any earlier Fibonacci number.<br /><br />\(144^5 = 27^5 + 84^5 + 110^5 + 133^5 \) This counter-example disproved Euler's Conjecture that n nth powers are needed to sum to an nth power. It is also part of one of the shortest papers ever published in a math journal(two sentences)</span><br /><div class="separator" style="clear: both; text-align: center;"><span style="font-family: inherit; font-size: small;"><a href="https://1.bp.blogspot.com/-Ex-N8ZqGiDI/VzOr1JnJirI/AAAAAAAAIPM/pw0T68jsq5wZOXJNVuNsMsT4A2pOv4uswCLcB/s1600/euler%2Bcounter%2Bexample%2Bshort%2Barticle.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="268" src="https://1.bp.blogspot.com/-Ex-N8ZqGiDI/VzOr1JnJirI/AAAAAAAAIPM/pw0T68jsq5wZOXJNVuNsMsT4A2pOv4uswCLcB/s640/euler%2Bcounter%2Bexample%2Bshort%2Barticle.jpg" width="640" /></a></span></div><span style="font-family: inherit; font-size: small;"><br /><br />(Squares are important in knowing if a number, n is Fibonacci or not. N is Fibonacci IFF one or both of \(5n^2 \pm 4\) is a perfect square. )<br /><br />144 is also the smallest square number which is also a square when its digits are reversed 144 = 12 <sup>2</sup> while 441= 21<sup>2</sup><br />144 is the second smallest even square which has no prime for its two adjacent numbers, 143 = 11 x 13, 145 = 5 x 29 . There is only one more year after this which is a square with no adjacent prime.</span></div><div><span style="font-family: inherit; font-size: small;"><br />The sum of the first 144 decimal digits of pi (don't use the 3.) is 666, "The Number of the Beast." One person wrote that they thought that was gross! ( sorryf :-{ , bad pun)<br /><br />144 is the only year day that is a square number that is the perimeter of a primitive Pythagorean Triangle. (16, 63, 65) *Ben Vitale<br /><br />Srinavas Raghava K gives four different expressions for 144 in different sets</span><br /><span style="font-family: inherit; font-size: small;">Using the "golden ratio", \(\phi^2 + \phi^6 + \phi^10 + \phi^{-2} + \phi^{-6} + \phi^{-10} \)</span><br /><span style="font-family: inherit; font-size: small;">With Trangular Numbers \( T_2 + T_8 + T_14 \)</span><br /><span style="font-family: inherit; font-size: small;">With Lucas Numbers \( L_2 + L_ 6 + L_10\) students should observe the simmularity of the exponents with the golden ratio example. </span><br /><span style="font-family: inherit; font-size: small;">With Fibonacci Numbers</span><br /><span style="font-family: inherit; font-size: small;">\(F_12\) = 144<br /><br />And from Das Ambigramm 144 = 2x2x3x3x2x2 </span><br /><span style="font-family: inherit; font-size: small;"><br /></span><span style="font-family: inherit; font-size: small;">1729, the famous Taxi-cab number, was known to Ramanujan because he was studying "Fermat near-misses", numbers where z^n was only one away from x^n + y^n for some x, y, z. In his case, 1729 was a cube that was one more than 9^3 + 10^3 = 1729, one more than 12^3. 144 is another such number, but seldom celebrated. 71^3 + 138^3 = 144^3 - 1. (there is one more smaller year day which also is a "Fermat near-miss" of this same kind, and it is related to 1729.) </span></div><div><span style="font-family: inherit; font-size: small;">And 144 appears in another cubic near-miss, 73^3 + 144^3 = 150^3 + 1</span></div><div><span style="font-family: inherit; font-size: small;"><br /> 144 is the smallest Fibonacci number in a string of five consecutive Fibonacci numbers that sum to a prime number. It is also the 2nd smallest in another such sequence.</span><br /><div><span style="font-family: inherit; font-size: small;">(and off the wall, did you ever notice if you take four consecutive Fibonacci numbers, A, B, C, D, then BC-AD = 1.)<br /><br />144 is the largest magnitude for the determinant of a 9x9 binary matrix. </span></div><div><br /></div><div>144 has a special link with the numbers 17 and 12, which appear in Theon's ladder of approximations to the square root of two. 144 is the 17th triangular number, and the 12th square number. Every other number in Theon's ladder (even denominators) gives a list of ALL the numbers which are both square and triangular by their indices. 144 is the largest year day that is both triangular and square, the next such number is from the ratio 41/29 (the 41st triangular number and the 29th square number, 841.<br /><hr /><span style="font-family: inherit; font-size: small;"><b>The 145th Day of the Year:</b> 145= 1! + 4! + 5!. There are only four such numbers in base ten. 1, 2 and 145 are three of them, what's the fourth? Such numbers are called factorions, a term created by Cliff Pickover in 1995<br /><br />145 is the result of 3<sup>4</sup> + 4<sup>3</sup>, making it a Leyland number. a number of the form x<sup>y</sup> + y<sup>x</sup> where x and y are integers greater than 1. They are named after the British number theorist, Paul Leyland. (There are ten days of the year that are Leyland numbers)<br /><br />Prime Curios points out several curiosities related to 145, The 145th prime number is 829 and their concatenation, 145829 is prime. And the largest prime factor of 145, is 1+4+5+8+2+9. and 149 is congruent to 1 in mod 8, mod 2, and mod 9.<br /><br />The process of summing the squares of the digits of a decimal number has two results, one is the eventual decent to 1, and being called a happy number. 145 is the largest Unhappy number. Unhappy numbers eventually land on one of the numbers in the eight cycle, 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 ... (any three digit number, for example, produces a sum of squares less than or equal to 243. Any of the numbers you land on that are greater than 145 and less than 243 has a sum of squares of its digits that is less than itself, and eventually they land on one of the chains that lead to the eight cycle shown. Some numbers (like 99) iterate to a number greater than 145, but they then recede back into the inexorable "cycle of unhappiness" above. (A great exploration for students to create the trees of all numbers less than 200 that go to either 1, or the unhappy cycle)</span></div><div><br /></div><div>The 145th prime number is 829, and 145829 is prime. Notice also, that the largest prime factor of 145 = 1+4+5+8+2+9, and that 145 is congruent to 1, mod 8, mod 2, and mod 9. *Prime Curios</div><div><br /></div><div>145 is a palindrome in base 12, (101)</div><div><br /></div><div>145 can be written as the sum of two squares in two different ways, 12<sup>2</sup>+1<sup>2</sup> = 8<sup>2</sup>+9<sup>2</sup><br /><hr /><b>The 146th Day of the Year</b><br /><br />146 is 222 in base eight. *What's So Special About This Number<br /><br />Jim Wilder@wilderlab pointed out that the sum of the divisors of 146; 1+2+73+146 also equals 222. Finding value of 222 in base n is nice introduction to polynomials, and (IMHO) leads students to understand polynomials (and base 10) much better.<br /><br />The decimal expansion of 1/293 has a period of 146 digits.<br /><br />146 is a number n, for which n<sup>2</sup>+1 is prime. Goldbach conjectured that any number in this sequence could be written as the sum of two other numbers in the sequence. For 146, one such solution is 146 = 20 + 126 *OEIS<br /><br />The absolute difference between any two digits of this number is prime. *Prime Curios, For how many three digit numbers is this true?<br /><br />Another nice palindrome from Das Ambigram, 146 = 2x5x7+3+3+7x5x2<br /><br />If you roll two pairs of standard fair dice, the number of ways that both pair can turn out with equal face value showing is 146 out of 1296. The numerator for getting any of the numbers 2 through 12, is an interesting sequence, 1 + 4 + 9 + 16 + 25 + 36 + 25 + 16 + 9 + 4 + 1.... Guess you could say all fair and SQUARE.<br /><br />146 is Roman Numerals uses all the symbols below 1100, CXLVI once each.<br /><hr /><b>The 147th Day of the Year</b><br /><br />if you iterate the process of summing the cubes of the digits of a number starting with 147, you eventually start repeating 153. This seems to be true for all multiples of three.<br /><br />Shorty palindrome from Das Ambigramm 147 = 7*3*7.<br />He also added that 147 = 4+5+6..... + 16 + 17 = 18+19+...+ 23+24.<br /><br />144 is the sum of two Fibonacci numbers, F(12) + F(4 )= 144 + 3 = 147</div><div><br /></div><div>If there are no fouls, the maximum score on a snooker break is 147.<br /><br />And Derek Orr@<b>Derektionary</b> pointed out that "147 is the smallest number formed by a column of numbers on a phone button pad"<br /><br />147 in binary has an equal number of zeros and ones.<br /><br />The binary form of 147 (10010011) contains all the two-digit binary numbers (00, 01, 10 and 11).<br /><br />147 is a repdigit in base 20 (77), or 7 score and 7.<br /><br />Not even sure if it is unusual, or how unusual, but 147 ends in a digit of three in bases 4, 6, 8, 9, 12, 16, and 36.<br /><hr /><b>The 148th Day of the Year</b><br />148 "<i>Primelicious</i>", 2<sup>1</sup> + 1 is prime,2<sup>4</sup> + 1 is prime, and 2<sup>8</sup> + 1, and the three results add to a prime, 3+17+257 = 277. Looking for more Primelicious numbers.<br /><br />148 is also a Loeschian number, a number of the form a<sup>2</sup> + ab + b<sup>2</sup>. These numbers and the triples (a,b,L) formed by points in space are used, among other places in locations of spheres under hexagonal packing. (These numbers are named after August Loesch, German Economist {1906-1945})<br /><br />A Vampire number is a number whose digits can be regrouped into two smaller numbers that multiply to make the original (1260 = 21*60). There are 148 vampire numbers with six digits. (There are 7 four-digit vampire numbers, which might be easier for younger students to find. (1260, 1395, 1435, 1530, 1827,and 2187)<br /><br />148 is the 12th number in the Mian-Chowla sequence. The sequence starts with 1, and numbers are added in order if they do not a sum of some distinct collection of existing numbers. So 2 is in, and we have 1, 2. Now 3 can't be added since 1+2 = 3, so we go to four, .... 1,2,4. Now 5, 6, and 7 can all be formed from those present, so 8 will be next. (The sequence is named for Sarvadaman Chowla, a British born Indian American Mathematician (1907-1995, one of the co-creators.)<br /><br />\(e^{\pi\sqrt{148}}\ is an integer..... almost, 39660184000219160.00096667...</div><div><br />148 is a Palindrome in base 6(404) and base 36 (44).<br /><hr /><b>The 149th Day of the Year</b><br />149 is the 35th prime number, and a twin prime with 151.<br /><br />145 is an Emirp since 941, its reversal, is also a prime.<br /><br />There are 149 ways to put 8 queens on a 7-by-7 chessboard so that each queen attacks exactly one other queen. *Prime Curios<br /><br />149 in binary is 10010101. The zeros are in prime positions 2, 3, 5, and 7, when read left-to-right. These are the four single digit prime numbers.*Prime Curios<br /><br />149 is a strictly non-palindromic number, it is not a palindrome in any base from 2 to 147.<br /><br />149 is a full reptend prime, its reciprocal is 148 digits long, 1/149 repeats 0067114093959731543624161073825503355704697986577181208053691275167785234899328859060402684563758389261744966442953020134228187919463087248322147651 indefinitely.<br /><br />also 149 = 6<sup>2</sup> + 7<sup>2</sup> + 8<sup>2</sup>.(note that the digits 1, 4, 9 are squares also)<br /><br />149 is also the sum of three primes, none of which are the sum of two squares, 23 + 43 + 83 And Derek Orr noted that the sum of the digits of 149, \(1 + 4 + 9 = 14 = 1^2 + 2^2 + 3^2 \)<br /><br />149 is the smallest 3-digit prime with distinct digits in each position such that inserting a zero between any two digits creates a new prime (that is, 1049 & 1409 are both prime).<br /><br /><hr />The 150th Day of the Year<br />150 is the largest gap between consecutive twin prime pairs less than a thousand. It occurs between {659, 661} and {809, 811}. *Prime Curios<br /><br />2<sup>150</sup> - 3 and <sup>150</sup> - 5 are twin primes. BIG twin primes! and one more from Prime Curios.... 150 is the largest gap between consecutive twin prime pairs less than a thousand. It occurs between {659, 661} and {809, 811}.<br /><br />150 is the sum of eight consecutive primes starting with 7.<br /><br />150 is a Harshad(joy-giver) number, divisible by the sum of its digits.<br /><br />A really nice sequence of infinite sums from Nakasu Wataru on Twitter,<br /><br />\( \sum{k=1}^{\infty}(\frac{k}{2^k} = 2\)<br /><br />\( \sum{k=1}^{\infty}(\frac{k^2}{2^k} = 6\)<br /><br />\( \sum{k=1}^{\infty}(\frac{k^3}{2^k} = 26\)<br /><br />\( \sum{k=1}^{\infty}(\frac{k^4}{2^k} = 150\)<br />And that's the last one that's a year day.<br /><br />150 is a palindrome in base 4(2112), and in base 7(303) A Poly divisible number is an n-digit number so that for the first digit is divisible by one, the first two digits are divisible by two, the first three digits are divisible by three, etc up to n. There are 150 three-digit poly divisible numbers. Hat tip to Derek Orr .<br /><br />150 = 5 x 2 x 5 + 5 x 2 x 5 + 5 x 2 x 5 HT to Das Ambigram<br /><br />150 year celebration is called sesquicentennial of the event.<br /><br />And... 150 is the number of degrees in the quincunx astrological aspect explored by Johannes Kepler.<br /><br />Rubix Cube gotten too easy for you? Try the Professor's Cube, 150 movable facets.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-5jTml1ol8tQ/XpjfTHEa1mI/AAAAAAAAL9Y/RGRRidXuvPUs3PA5Ck0B5pevRyL7gpZxwCLcBGAsYHQ/s1600/Proffessors%2Bcube.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="220" data-original-width="220" height="320" src="https://4.bp.blogspot.com/-5jTml1ol8tQ/XpjfTHEa1mI/AAAAAAAAL9Y/RGRRidXuvPUs3PA5Ck0B5pevRyL7gpZxwCLcBGAsYHQ/s320/Proffessors%2Bcube.jpg" width="320" /></a></div><hr /><br /></div></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-44765650700391653182020-03-21T12:25:00.028-07:002021-04-21T11:01:30.840-07:00Number Facts for Every Year Day (91-120)<b>The 91st Day of the Year:</b><br />The 91st day of the year; 10<sup>n</sup> + 91 and 10<sup>n</sup> + 93 are twin primes for n = 1, 2, 3 and 4. (For bases less than ten, one of these expressions is prime for some other values of b^n, which?)<br /><br />91 and it's reversal 19 are related to Ramanujan's Taxi-cab number, 1729 = 19x91, a palindrome product. Note that the sum of the digits of 1729 are 19.<br /><br />91 is : The sum of thirteen consecutive integers = 1 + 2 + 3 + ... + 11 + 12 + 13, and thus the thirteenth triangular number.<br />and of six consecutive squares= 1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> making it a pyramidal number,<div><br /><div>91 is the sum of two consecutive cubes = 3<sup>3</sup> + 4<sup>3</sup> and the difference of two consecutive cubes = 6<sup>3</sup> - 5<sup>3</sup><br /><br />91 is also the sum of three squares, 1^2 + 3^2 + 9^2 .<div><br /></div><div>91 is the sum of the first three powers of 9 starting with 0, \( 9^0 + 9^1 + 9^2) = 91, the 13th triangular number. Every number produced the the sequence of powers of nine is a triangular number 9^0 = 1, 9^0 + 9^1 = 10, etc. Taken to the third power you get 820 which is t(40), and for the fourth power you get 7381 which is t(121)</div><div><br /></div><div>91 = 46^2 - 45^2 = 10^2 - 3^2</div><div><br /><a href="https://1.bp.blogspot.com/-a3vTXWFduhQ/XlQCZuWp2QI/AAAAAAAALUI/9E5lqjhHmCIMl1zAP4d4rwGjpDuC3xMwgCLcBGAsYHQ/s1600/half%2Bcent.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="77" data-original-width="77" src="https://1.bp.blogspot.com/-a3vTXWFduhQ/XlQCZuWp2QI/AAAAAAAALUI/9E5lqjhHmCIMl1zAP4d4rwGjpDuC3xMwgCLcBGAsYHQ/s1600/half%2Bcent.jpg" /></a>The sum of one of each US coin less than a Silver Dollar is 91 cents. (actually, if you accept some very old US coins, there was once a 5 mil coin, adding 1/2 cent to this total. <b><br /></b><br />91 is the smallest non-trivial odd composite (that is, its prime factors [7, 13] are not, at first glance, obvious). Every smaller odd composite is either a familiar square, ends in 5, has a digit sum that is a multiple of 3, or is obviously divisible by 11. *Prime Curios<br /><br />91 is a repdigit in base 9 (111) (<i>9^2+9+1</i>) and a palindrome in base 9 and base 3 (10101) (3^4 + 3^2 + 1)<br /><br />91 is the smallest pseudoprime (two unique prime factors) for which it is true that 3<sup>n</sup> = 3 mod n *Prime Curios<br /><br />Prime numbers less than 10,000,000 occur with the two digit ending 91 more than any other ending. *Prime Curios<br /><hr /><b>The 92nd Day of the Year:</b><br />The 92nd day of the year; 92 is the smallest composite number for which the reverse of its digits in hexadecimal, decimal, octal, and binary are all prime. *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios </a> for instance in base 8 it is expressed as \(431_8 \) and it's reversal, \(134_8 \) =89<br /><br /> And... There are exactly 92 Johnson Solids: The Johnson solids are the convex polyhedra having regular faces and equal edge lengths (with the exception of the completely regular Platonic solids, the "semiregular" Archimedean solids, and the two infinite families of prisms and antiprisms). *Geometry Fact @GeometryFact and a related point, The snub dodecahedron has 92 faces (80 triangular, 12 pentagonal), the most any Archimedean solid can have.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-WH5_h8gNdwY/VvRjRRJI1EI/AAAAAAAAIIc/SJeLYeCo2dMuCJvMqN6PZpqk1ToS0qucA/s1600/snub%2Bdodecahedron.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-WH5_h8gNdwY/VvRjRRJI1EI/AAAAAAAAIIc/SJeLYeCo2dMuCJvMqN6PZpqk1ToS0qucA/s1600/snub%2Bdodecahedron.gif" /></a></div>92 is the number of different arrangements of 8 non-attacking Queens on an 8 by 8 chessboard (i.e. no two Queens should share the same row, column, or diagonal)<br /><br /> 92= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 the sum of eight consecutive integers (when you cut off the top of an equilateral triangle with a cut parallel to the base, the remaining quadrilateral is an isosceles trapezoid, so why not call these cut off triangular numbers, trapezoidal numbers)<br /><br />92 is a palindrome in bases 6 (232<sub>6</sub>), and 7 (161<sub>7</sub>) <br /><br />Unlike 91 (and lots of other numbers) 92 can not be written as the sum of three positive squares.</div><div><br /></div><div>Because 92 is divisible by four, it is the difference of two squares, 24^2 - 22^2</div><div><hr /><b>The 93rd Day of the Year:</b><br />The first 93 digits of 93! form a prime number. *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios </a><br /><pre>93! = 1156772507081641574759205162306240436214753229576413535186142281213246807121467</pre><pre>315215203289516844845303838996289 ...</pre><pre><br /></pre><pre><span style="font-family: "Times New Roman"; white-space: normal;">93 is a Plum Integer since both its divisors, 3 and 31, are Gaussian Primes (Primes of the form 4n+3)</span></pre><pre></pre>93 is the sum of three distinct squares, 93 = 2<sup>2</sup> + 5<sup>2</sup>+ 8<sup>2</sup> and six consecutive integers 93= 13 + 14 + 15 + 16 + 17 + 18 (another trapezoidal number, see (92nd Day)<br /><br />There are 93 five-digit prime palindromes. The smallest (I think) is 10301<br /><br />A potato can be cut into 93 pieces with just nine straight cuts.<br /><br />and 93 in base 10 is 333 in base 5<br /><br />93, 94, and 95 form the third string of three consecutive semiprimes (two distinct factors)<br /><br />93 is a palindrome in base 2, \( 1011101_2\) and in base 5 \( 333_5\)<br /><br />There are 93 different real periodic points of order 11 on the Mandelbrot set.<br />2<sup>60</sup> - 93 is prime. The statement is untrue if 93 is replace with any integer smaller than 93 <b><br /></b><br />93 = 47^2 - 46^2 = 17^2 - 14^2<br /><hr /><b> The 94th Day of the Year</b><br />94!-1 is prime. The number 94!-1 ends in 21 consecutive nines. Students might inquire how they could have known this without being told.<br /><br />94 begins the smallest string of three consecutive numbers none of which is a palindrome in any base, b \( 2 \leq b \leq 10 \)<br /><br />Add the prime factors of 94 and the result is 49, 94 reversed.<br /><br />The sum of digits of the distinct prime factors of 94 add up to 13, which is also the sum of the digit of 94. 94 = 2 x 47 and 2 + 4 + 7 = 13. Such numbers are called Hoax numbers. 94 is also a Smith number, which is the sum of the digits of all prime factors, including multiplicity, (see day 364 for more)</div><div><br />94 is the smallest even number greater than four which cannot be written as a sum of two twin primes. *Prime Curios<br /><br />94 has all square digits, The 94th prime is 491, also with all square digits.<br /><br />93, 94, and 95 form the third string of three consecutive semiprimes (two distinct factors) In all the numbers up to 10^9, the longest string of semiprimes is 94.<br /><br />and 94 is the 29th semiprime, and the fourteenth with 2 as one of those two prime factors.<br /><br />1100977 and 1101071 are a pair of consecutive primes. They form the last pair of primes known that are two digits (94) apart.<br /><br />94 is the smallest number above the trivial number 1, that is equal to the sum of the squares of its digits in base 11. 94<sup>2</sup>= \$ 8836_{10} = 673_{11}\$ and \$ 6^2 + 7^2 + 3^2 = 94 \$ *Wik<br /><br /></div><div>Most mathematicians know the story of 1729, the taxicab number which Ramanujan recognized as a cube that was one more than the sum of two cubes, or the smallest number that could be expressed as the sum of two cubes in two different ways. But not many know that 103 is part of the second such \(64^3 + 94^3 = 103^3 + 1^3 \)<br /><b></b><br /><hr /><b>The 95th Day of the Year:</b><br />95<sup>0</sup> + 95<sup>1</sup> + 95<sup>2</sup> + 95<sup>3</sup> + 95<sup>4</sup> + 95<sup>5</sup> + 95<sup>6</sup> is prime. *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a><br /><br />95 and its reversal (59) begin fewer four-digit prime numbers (seven) than any other two-digit number.<br /><br />95 is the number of planar partitions of 10. (A plane partition is a two-dimensional array of integers n_(i,j) that are nonincreasing both from left to right and top to bottom and that add up to a given number n. Here's some different plane partitions of the number 10 and of course all of them could go vertically as well. 5 2 2 1<br />4 2 2 2<br />3 2 2 2 1<br /><br /><br /> 95 is the sum of 7 consecutive primes = 5 + 7 + 11 + 13 + 17 + 19 + 23<br /><br />NINETYFIVE is the largest semiprime that can be spelled with a semiprime number of toothpicks. *Prime Curios<br /><br />95 is the third member of the third sequence of three consecutive semiprimes. <br /><br />Magic Johnson once got 75 assists in a 7 game NBA playoff series. Still the record at the time that I write this. And strangely coincidental is that 95 is also the largest number of free throws ever attempted in one 7 game NBA playoff series, by another Laker, Jerry West.... not a good guy to send to the line. (would love to know if this were both the same 7 game series.) <br /><hr /><b>The 96th Day of the Year</b>:<br /><br />96 is the smallest number that can be written as the difference of 2 squares in 4 ways. *<a href="http://www2.stetson.edu/~efriedma/numbers.html" target="_blank">What's So Special About This Number? </a>(<i>students are encouraged to find them all</i>...Is there a smaller number that can be so expressed in 3 ways?)<br /><br />96 is the smallest natural number whose factorial begins with the digit nine, it has 150 digits. Students should be able to know how many zeros are on the end of that 150 digit string<br /><br />96 is a strobogrammatic number, rotated 180 degrees it is still 96.<br /><br />The sum of 96 consecutive squared integers is a square number ( \$ x^2 + (x+1)^2 + (x+2)^2 +(x+3)^2 + \dotsm + (x+95)^2 = y^2 \$ ) can be solved with eight sets of 96 consecutive year days. One solution is \$ 13^2 + 14^2 + \dotsm + 108^2 = 652^2 \$ *Ben Vitale<br /><br />Ninety Six, South Carolina. <span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">Ninety Six figured prominently in the </span>Anglo-Cherokee War<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> (1758–1761). During the </span>American Revolutionary War<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, it was a site for </span>southern campaigns<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">. The first land battle of the revolution south of </span>New England There is much confusion about the mysterious name, "Ninety-Six," and the true origin may never be known. Speculation has led to the mistaken belief that it was 96 miles to the nearest Cherokee settlement of Keowee; to a counting of creeks crossing the main road leading from Lexington, SC, to Ninety-Six; to an interpretation of a Welsh expression, "nant-sych," meaning "dry gulch." Pitcher Bill Voiselle of the Boston Braves was from Ninety Six, South Carolina, and wore uniform number 96.<br /><br />\$ \Pi (96) = \frac{96}{4} \$ (The number of primes less than 96 is equal to 96/4) It is the smallest Year Day for which this is true.<br /><br />Superprime numbers are prime numbers whose prime index is also a prime number. For example 5 is prime, and it is the third prime, so it is a superprime. Every integer greater than 96 can be represented as the sum of distinct superprimes.<br /><br /><hr /><b>The 97th Day of the Year:</b><br />The number formed by the concatenation of odd numbers from one to 97 is prime. (1+3+5+7+9+11+13+15+17+... 93+95+97 quick students, how many digits will it have?) *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a><br /><br />The sum of the first 20 digits of pi is 97. If you add the next digit, you get another prime, 103. There are only 11 prime year days that are the sum of the first n digits of pi. These two are the fifth and sixth of them. 313 is the 11th and largest of them, and is the sum of the first 63 digits of pi. </div><div><br /> And from Cliff Pickover, 97 is the largest prime that we can ever find that is less than the sum of square of its digits 9<sup>2</sup> + 7<sup>2</sup> > 97<br /><br />There are 97 leap days every 400 years in the Gregorian Calendar<br /><br />The smallest prime which is the sum of a prime number of consecutive primes as well as the sum of a composite number of consecutive composite numbers: 97 = 29 + 31 + 37 = 22 + 24 + 25 + 26. *Prime Curios <br /><br />The smallest prime that has a prime alphabetic value in its Roman numerals based representation, i.e., XCVII -> 24 + 3 + 22 + 9 + 9 = 67. <br /><br />The reciprocal of 1/97 begins with powers of three, .010327835... in two digit brackets. So what happened to 81? Try looking at the powers of three spaced two apart in a column <br />01<br /> 03<br /> 27<br /> 81<br /> 243<br /> 729....<br />______________________<br /><br />Now add<br /> look at 1/997.<br /><br />If your watching your weight, a jigger (1.5 oz) of most spirits contain about 97 calories. Please blame *Prime Curios<br /><br />between 100 and 1000, there are 97 primes with distinct digits.<br /><br />For the four fours game, 97 = 4! * 4 + (4/4). Can you do it with my five factorials game using 1!, 2!, 3!, 4!, and 5! and only +,-,*,/ operations<br /><br />The longest whole-number name consisting entirely of alternating consonants and vowels is NINETY-SEVEN. However, if all integers are allowed, NEGATIVE NINETY-SEVEN would qualify.<br /><br />\$97 = 2^4 + 3^4 \$ the sum of two consecutive primes to the same power. It is the largest known prime with this property.. <br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> 97, 907, 9007, 90007 and 900007 are all primes.</span><br /><hr /><b>The 98th Day of the Year:</b><br />98 is the smallest number that starts a sequence of three consecutive numbers with at least 3 prime divisors. (<i>What would be the smallest number to start a sequence of four numbers with at least four prime divisors?</i>)<br /><br />98 is the sum of fourth powers of the first three integers, 1<sup>4</sup> + 2<sup>4</sup> + 3<sup>4</sup> Only one larger year day is the sum of the first 3 nth powers .<br /><br />98 is the smallest composite number whose reversal, 89, is a Fibonacci prime. (is there a reversible composite that is prime but not a Fibonacci number, or a Fibonacci number but not prime?)<br /><br />98 is a ambinumeral, rotating it 180 degrees produces another integer, 86.<br /><br />98 is a palindrome in base 5 (343) , and base 6 (242)</div><div><br /></div><div>If you take a number and add it to its reversal, such as 104 + 401 = 505, you get a palindrome. And if you don't, just repeat the process. 75+57 = 132, and 132 + 231= 333. If you try this process with 97, be patient. It takes 24 steps to get a palindrome.... but you do get a palindrome. <br /><hr /><b>The 99th Day of the Year:</b><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">If 99 divides some 4-digit number ABCD, then 99 also divides BCDA, CDAB, and DABC</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">There are 9 ways to express 99 as p + 2q, where p and q are prime. (Students might wonder why this strange p+2q idea should be interesting. It</span><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; text-indent: -1em;"> is related to a conjecture of Lemoine. </span><br /><div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; margin-left: 1em; overflow-wrap: break-word; text-indent: -1em;"><tt>The conjecture states that any odd number greater than 5 can be written as <b>p</b>+<b>2q</b> where p and q are primes. Students might try to find the several numbers smaller than 99 that can be expressed in p+2q form over 10 ways.) </tt></div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">99 is the largest number that is equal to the sum of its digits plus the product of its digits: 99 = 9 + 9 + 9 * 9</span><br /><br />99 is the sum of the cubes of three consecutive numbers, \( 2^3 + 3^3 + 4^3 \)<br /><br />99 is the sum of all the sums of all the divisors (including themselves) from one to 11.*Wik<br /><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">and 99 is the alphanumeric value of THIRTEEN *</span><a href="http://numbergossip.com/" style="background-color: white; color: #888888; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-decoration-line: none;" target="_blank">Number Gossip</a><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"></span><br /><br />99 is a palindrome in base two, and quite a pretty one, (1100011<sub>2</sub>) as well as in base ten, where it is a rebdigit also.<br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">99</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"> = 9801 and 98 + 01 = 99 so it is a Kaprekar number, named after D. R. Kaprekar, an Indian recreational mathematician.</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">and David Marain @dmarain recently reminded me 1/99</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"> = It is the eleventh Year Day that has this quality, but there are only 9 more for the rest of the year. </span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"><br /></span><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">0.000102030405060708091011121314151617181920212223242526272... The question for students, It must be a repeating decimal, when does it start to repeat?</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"><span style="font-size: medium;">(and there was something about bottles of beer on the wall, but they don't seem to be there anymore. Maybe someone took them down...)</span></span></div><div><span style="color: #222222; font-size: medium;"><br /></span></div><div><span style="color: #222222; font-size: medium;">All fifty of the odd numbers up to ninety-nine can be arranged with the first 25 summed in the numerator, and the second 25 in the denominator, and the result is 1/3. But you can do that with all the odds from 1 to 4n-1 for any n. Here is a beautiful proof without words from the brilliant @Futility Closet and credited to Roger Nelson. And a HT to @mathhombre for the heads-up. </span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tTrpaS6l9ow/YFOeI1y9ZhI/AAAAAAAANbA/vUvw5Y1jD-cXmvA3jYnc2kOfkxkM4iaMQCLcBGAsYHQ/s900/odds%2Bmake%2Ba%2Bthird.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="688" data-original-width="900" src="https://1.bp.blogspot.com/-tTrpaS6l9ow/YFOeI1y9ZhI/AAAAAAAANbA/vUvw5Y1jD-cXmvA3jYnc2kOfkxkM4iaMQCLcBGAsYHQ/s320/odds%2Bmake%2Ba%2Bthird.png" width="320" /></a></div><br /><span style="color: #222222; font-size: medium;"><br /></span><br /><hr /><b>The 100th Day of the Year: </b><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">The first 3 primes add to 10 and the first 3</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"> primes add to 10</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;">2</sup><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"> = 100 *</span><a href="http://primes.utm.edu/curios/home.php" style="background-color: white; color: #888888; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-decoration-line: none;" target="_blank">Prime Curios</a><br />and the cube of the first four positive integers sum to 100, 1³ + 2³ + 3³ + 4³ *Jim Wilder</div><div><br /></div><div>And Hansrudi Widmer tweeted that 100 = 2⁶ + 6². </div><div><br /></div><div><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">And 100=1+2+3+4+5+6+7+(8•9) *jim wilder @wilderlab or 123 + 4 - 5 + 67 - 89 = 100 *Alexander Bogomolny @CutTheKnotMath There are many more of these, find your own. Using only + or - there is only one way using exactly 7 +/- signs. This classic old problem is generally credited to Henry Ernest Dudeney whose birthday is today (see below) .</span><br />Hey! Can you make 100 in my Five Factorials Game. Use 1!, 2!, 3!, 4!, 5! and only the operations +, -, *, and /<br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">The last proof in John Horton Conway's "On Numbers and Games" is: Theorem 100; "This is the last Theorem in this book.The Proof is Obvious."</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">How many legs does a centipede have? Although the name is derived from cent(100) and ped (foot) the answer is NOT 100! In fact, it seems that all centipedes have twice an odd number for the number of legs so they can't have 100. In "The Book of General Ignorance" it is said that one (or at lest one) variety of centipede had been found with 96 legs, this seems not to be supported by the folks who study the creatures. There are some types that seem to have 2*49 = 98 legs, but none have been found with 100 legs (and none are expected to be found)</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">West Virginia seems to have more communities with numerical names than anywhere else in the world. They have a Six, and an Eight, and they even have the only town in the US named Hundred. Originally named "Old Hundred" for a long lived early settler, Henry Church. The sign points out that Henry served for the British in the Revolutionary War, but doesn't include that he took up arms to fight against them in the War of 1812. Before he arrived at his assignment, the war ended, so he returned to his home in Hundred.</span><br /><div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: center;"><a href="https://1.bp.blogspot.com/-bk2E7XAP-yU/Vs8tbRxnp1I/AAAAAAAAH5g/Pd3esNAYLIY/s1600/HundredWVASign.jpg" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" src="https://1.bp.blogspot.com/-bk2E7XAP-yU/Vs8tbRxnp1I/AAAAAAAAH5g/Pd3esNAYLIY/s1600/HundredWVASign.jpg" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" /></a></div><br />100 is the last year date, which can be expressed as consecutive triangular numbers in more than one way. \(T_5 + T_6 + T_7 + T_8 = T_9 + T_10\) . That's 15 + 21 + 28 + 36 = 45 + 55. <br /><hr /><b>The 101st Day of the Year:</b><br />101 is a self-strobogramatic number. It is the second smallest Prime self-strobogrammatic prime, after 11<br /><br />There are six ways you can pick two of the four smallest primes, 2, 3, 5, and 7. Form all six pairs, multiply each pair, and add all the products....boom, 101<br /><br />3! - 2! + 1! = 5 (prime) 4! - 3! + 2! - 1! = 19 (prime) 5! - 4! + 3! - 2! + 1! = 101 (prime) HT to Ed Southal (<i>What would be the next number created in a sequence like this? Is it prime?</i>)<br /><br />101 is the sum of five consecutive primes, It is the fourth prime year day that is the sum of five consecutive primes.<br /><br /> 101 = 5! - 4! + 3! - 2! + 1!<br /><br />After lunch, try to be talking to someone else near a digital clock, then when 1:01 clicks up, you can point and say, "Ahh, that's the smallest prime number you will ever see on a clock." If they try to be clever and ask what's the next, pause and say, "Ummm, give me two minutes."<br /><br />101 is the largest known prime of the form 10<sup>n </sup>+ 1.<br /><br /> There are 101 digits in the product of the 39 successive primes produced by the formula n<sup>2</sup> + n + 41, where n = 1 to 39. This formula was used by Charles Babbage to demonstrate the capabilities of his Difference Engine (1819-1822). *Prime Curios<br /><br /> and The last five digits of 101<sup>101</sup> are 10101.And.. the 101 Fibonacci number ends in 101, also.<br /><br />1 + 6 + 8 = 15 = 2 + 4 + 9, and the sets remain equal if you square them before adding, 1^2 + 6^2 + 8^2 = 2^2 + 4^2 + 9^2 = 101<br />Folks in Kentucky know that <span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"> Wild Turkey bourbon's most common production is its 101 proof. Brewed just down the road in Lawrenceburgh, Ky . Jump on the Bourbon Tour and stop by, and tell 'em Pat B sent ya'.</span><br /><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">Take any four digit repdigit, like 7777, and hey, its largest prime divisor is 101. *Prime Curios</span><br /><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">The sum of the squares of all the prime numbers up to 101 is prime.... </span><br /><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">and 101 is a palindrome in base ten, but not in any smaller base. </span><br /><br />Four of the X01 numbers are prime, 101, 401, 601,and 701 Of the other five, three are divisible by three, 301 has a smallest prime factor of 7, and 901 has a smallest prime factor of 17.</div><div><br /></div><div>Research into possible odd perfect numbers has revealed that the largest factors must be greater than 101, 10007, and 10000007. *Wolfram MathWorld <span face="Arial, Helvetica, sans-serif" style="background-color: white; font-size: 12px;"> </span><br /><hr /><b>The 102nd Day of the Year </b><br />I wrote that the number 102 may be the most singularly uninteresting number so far this year, but was corrected. Within an hour David Brooks sent me a list of items about 102.<br /> I really liked, and don't know how I missed, that "The sum of the cubes of the first 102 prime numbers is a prime number." Thanks David. It might be interesting for students to examine for which n is the sum of the cubes of the first n numbers (if any) a prime.)<br /> He also included that 102 is the name of a river in the state of Missouri. To French explorers the native American name for the river sounded like cent deux, the French words for 102. ( It is near the Iowa border, a tributary of the Platte River of Missouri that is approximately 80 miles long)<br /><br />Another writer wrote to tell me that 102 is the sum of four consecutive primes, 102 = 19 + 23 + 29 + 31,<br />And there used to be a US 102 in Michigan, but they did away with the name, and the road is now US 142.<br /><br /><hr /><b>The 103rd Day of the Year</b><br />there are 103 geometrical forms of magic knight's tour of the chessboard. <br /><br />103 is the reverse of 301. The same is true of their squares: 103<sup>2</sup> = 10609 and 301<sup>2</sup> = 90601. *Jim Wilder<br /><br /> The smallest prime whose reciprocal contains a period that is exactly 1/3 of the maximum length. (The period of the reciprocal of a prime p is always a divisor of p-1, so for 103 the period is 102/3 = 34. )<br /><br /> Using a standard dartboard, 103 is the smallest possible prime that cannot be scored with two darts. <br /><br />If you concatenate the numbers from 103 down to one (10310210101100....) the result is divisible by 101<br />If you take all the two digit prime numbers, and add up only their last digit, you get 103.... Think about the mind that searches that out. *Prime Curios<br /><br />103 is the smallest prime that doesn't appear in the first 3000 digits of Pi, 101 occurs in the first thousand, 107 appears in the second thousand,<br /><br />In yesterdays notes I pointed out that US 102 no longer exists. As far as I can find, US 103 never did. ANYONE?<br /><br />Most mathematicians know the story of 1729, the taxicab number which Ramanujan recognized as a cube that was one more than the sum of two cubes, or the smallest number that could be expressed as the sum of two cubes in two different ways. But not many know that 103 is part of the second such \(64^3 + 94^3 = 103^3 + 1^3 \)<br /><hr /><b>The 104th Day of the Year</b><br />104 is the smallest known number of unit line segments that can exist in the plane, 4 touching at every vertex. *<a href="http://www2.stetson.edu/~efriedma/numbers.html" target="_blank">What's Special About This Number</a><br /><br />104 is the sum of eight consecutive even numbers, 104 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20<br /><br />The reversal of 104 is a prime. It is the largest year day that has a prime reversal that is too large to be a year date<br /><br />13 straight lines through an annulus can produce a maximum of 104 pieces (students might try to create the maximum for smaller numbers of lines, the sequence is 2, 5, 9, 14, 20,... https://oeis.org/A000096 the differences give a clue to the complete pattern.)<br /><br />104 is a palindrome in base five (404), and in base six (252), and base twelve (88)<br /><br />and look at your keyboard, the standard Windows keyboard has 104 keys<br /><br />Japanese Route 104 ran from Hachinohe, near my former home in Misawa, Japan, on the Pacific to go across the mountains to Noshiro on the Sea of Japan in Akita prefecture. One of the better places to find the prized 36 inch green glass fishing floats washed up along the coast.<br /><br /><hr /><b>The 105th Day of the Year</b><br />Paul Erdős conjectured that this is the largest number n such that the positive values of n - 2<sup>k</sup> are all prime. *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a> Can you find a smaller number for which this is true?<br /><br />105 is the sum of consecutive integers in seven distinct ways. 105 = 1 + 2 + 3 + … + 13 + 14 = 6 + 7 + 8 + … + 14 + 15 = 12 + 13 + … + 17 + 18 = 15 + 16 + 17 + 18 + 19 + 20 = 19 + 20 + 21 + 22 + 23 = 34 + 35 + 36 = 52 + 53 <br /><br />As the sum of the first fourteen integers, it is a Triangular number.<br /><br />It is also the product of three consecutive primes, 3 x 5 x 7 = 105. Find the one smaller such year day, and the next larger.<br /><br />105 is the middle number in a prime quadruplet (101, 103, 107, 109) all in the same decade of numbers so it is the only odd composite in that decade of numbers. 15 holds a similar position in the teens decade.<br /><br />105 is the largest number for which all the odd composite numbers less than it,either share a factor with it, or are prime.<br /><br />105 is a palindrome in base four (1221) and in base eight (151) and base twenty (55).<br /><br />The distinct prime factors of 105, (3,5,7) add up to 15. The same is true of the factors of 104, so they form a Ruth Aaron pair. Someone noticed the factor relation about these two shortly after Hank Aaron hit his 715th home run to break Ruth's record of 714 on April 8th, 1974. 104 and 105 form the fifth such pair in year days, and yet, there is only one more for the rest of the year. <br /><hr /><b>The 106th Day of the Year</b><br />The sum of the first 106 digits of pi is prime. Amazingly, I could use this same numerical idea for tomorrow. And the sum of the first 106 digits, is prime also.<br /><br />106<sup>106</sup>-105<sup>105</sup> (a number of 215 decimal digits)is prime.<br /><br /> There are 106 distinct mathematical trees with ten vertices.<br /><br />106 is the fifteenth day of the year that in the form 2P where P is a prime.<br /><br /> Hundred, West Virginia was named for Henry Church and his wife, the first settlers who lived to be 109 and 106. Hundred is the only place in the United States with this name. <br /><br /><a href="https://1.bp.blogspot.com/-PLgmk6j0570/Xm9-L7ZLSAI/AAAAAAAALjo/SwFs43cSFFQ2XL299Hmh0gsvElU_jaWoACLcBGAsYHQ/s1600/Hell%2Bstore.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="216" data-original-width="170" src="https://1.bp.blogspot.com/-PLgmk6j0570/Xm9-L7ZLSAI/AAAAAAAALjo/SwFs43cSFFQ2XL299Hmh0gsvElU_jaWoACLcBGAsYHQ/s1600/Hell%2Bstore.jpg" /></a>106 is an invertible number, or strobogram. Some prefer to limit strobogram to only numbers that are themselves when rotated 180 degrees. Beyond the one digit numbers it is the smallest invertible semiprime (or biprime). 901= 17*53, is also a semiprime.<br /><br />M 106 in Michigan runs almost to Hell, literally, ending on M-36 just a few miles northwest of Hell, Michigan in the Pinckney State Recreational Area. If you came this far, you might as well stop by "Hell in a Handbasket Country Store", which used to be the Post Office for Hell, but now mail is delivered from Pinckney. Plan ahead, you might want to be there for Hellfest. They have an auto show, but only for hearses, and are in the Book of World Records for the longest Hearse parade in the world.</div><div><br /></div><div>106 and 107 are the second consecutive pair of n, n+1 such that the sum of the digits of Pi up to each is a Prime number. 106 is prime, and as mentioned below, the 107th digit of Pi is zero, so this is the pair of consecutive numbers in the sum of the digits of Pi where two identical primes occur. <br /><hr /><b>The 107th Day of the Year</b><br /><b><br /></b>There is no integer N such that N! has exactly 107 zeros in it. The same is true if we replace 107 by the primes 3, 31, or 43.*<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a> (<i>This seems a most remarkable set of facts to me.</i>)<br /><br />Interestingly, the sum of the first 107 digits of pi is prime, and the sum of the first 107 digits of e is prime. This is trivially true for the first digit of each, but can you find the one (I believe) other number between 1 and 107 for which the sum of the digits of e and pi are both prime?<br /><br />2<sup>107</sup> - 1 is the largest known Mersenne prime not containing all the individual digits. This number is a 33 digits long; <span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">162259276829213363391578010288127. . </span><br /><br /> Allan Brady proved in 1983 that the maximal number of steps that a four-state Turing machine can make on an initially blank tape before eventually halting is 107. <br /><br />If you remember AM radio, (OK, I know it's still out there) there are 107 possible carrier frequencies, in 10 Kilohertz bands from 505 to 1635 Kilohertz. *Prime Curios<br /><br />107 remains prime when 2 is added to any one of its digits. 307. 127, and 109 are all prime.<br /><br />107 is the sum of three prime numbers that all end in a 9. 19 + 29 + 59 = 107<br /><br /><br />107 +n! is prime for all single digits greater than one.<br /><br />107 and its reversal, 701 are both prime. Such pairs are generally called emirps<br /><br />And there is no smaller prime, p, for which the pth digit of Pi is zero. (If I counted right, the smallest odd number for which this is true is 77.) Zeros are amazingly scare in the first fifty digits, with only digits 32 and 50 being zero.</div><div><br /></div><div>106 and 107 are the second consecutive pair of n, n+1 such that the sum of the digits of Pi up to each is a Prime number. 106 is prime, and as mentioned above the 107th digit of Pi is zero, so this is the pair of consecutive numbers in the sum of the digits of Pi where two identical primes occur. <br /><hr /><b>The 108th Day of the Year</b><br />For fans of Pentominoes, you may try to construct all the Heptominoes, (made of 7 squares). There are 108 of them.<br /><br />108 can be written as the sum of a cube and a square (a^3 + b^2) in two ways. This is the smallest number with this property. *Prime Curios<br /><br />AND 108 = 1¹ • 2² • 3³ *jim wilder @wilderlab Numbers like this are called Hyperfactorials, this one is the third.<br /><br />The concatenation of 108 with its previous and next number is prime, i.e., 108107 and 108109 are primes.<br /><br />108 is the smallest possible sum for a set of six distinct primes such that the sum of any five is prime: {5, 7, 11, 19, 29, 37}. (Don't just sit there, there must be another that is larger. Find it.)<br /><br />108 is also, as every good geometry student knows, the interior angle measures of each angle of a regular pentagon</div><div><br /></div><div><span face="Roboto, sans-serif" style="background-color: white; color: #111111;">According to Vedic cosmology, 108 is the basis of creation, represents </span><span face="Roboto, sans-serif" style="background-color: white; color: #111111; font-weight: 700;">the universe and all our existence</span><span face="Roboto, sans-serif" style="background-color: white; color: #111111;">. In Hindu tradition, the Mukhya Shivaganas (attendants of Shiva) are 108 in number, and hence Shaiva religions, particularly Lingayats, use malas of 108 beads for prayer and meditation. And there are 54 letters in the Sanskrit alphabet, each appearing in masculine and feminine form, or 108 forms in all. HT Deb Jyoti Mitra</span><br /><br />Today and tomorrow are both examples of ambinumerals,or invertibl numbers, which form a different number when rotated 180<sup>o</sup> 108 becomes 801. Numerals like 181 which stay the same when rotated are called strobogrammatic numerals. This distinction is not always maintained.<br /><br /><b>The 109th Day of the Year,</b><br /><br />109 is the 29th prime, and the tenth superprime, a prime number whose prime rank, or index,(29) is also prime.<br /><br />On an infinite chessboard, the knight can reach 109 of them in three moves. (Just wondered how many a knight could reach on an infinite 3D chessboard... ANYONE?)<br /><br />M 109 in Michigan is called Dunes Hwy. and passes through the Sleeping Bear Dunes to connect Glen Arbor, Mi. A beautiful little village on the shore of Lake Michigan.<br /><br />109 is a twin prime with 107 and the largest of a prime quadruple including 103 and 101. I just found out that the product of twin primes (greater than 5) will have a digit root of 8..<br />5 * 7 = 35, 3 + 5 = 8<br />11 * 13 = 143, 1 + 4 + 3 = 8<br />17 * 19 = 323, 3 + 2 + 3 = 8<br />Hat tip to Ben Vitale<br /><br />109 = 1*2+3*4+5*6+7*8+9.<br /><br />109 is the smallest prime which is half the difference of two cubes, (7^3-5^3)/2<br /><br />The period of the reciprocal of 109 ends with 853211 (the beginning of the Fibonacci sequence reversed).<br />0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211....00917...<br /><br />109 rotated 180<sup>o</sup> is read as 601. I have enlisted the term ambinumerals for such pairs which are sometimes called invertible numbers. Numbers like 111 which are the same under rotation are known as strobograms. Some folks use strobogram for all of them, suitable since the root refers to spinning, not equality. In Roman numerals, CIX, it is its own reflection in a horizontal line. And 109 and 601 are the only ambinumeral pair that show up on a digital clock.<br /><br />109 is the smallest mumber which has more different digits than it's square. 109^2 = 11881, and you can partition 109 into 1 + 29 + 50 + 29. Concatenate those numbers in the order they are written and you get 1295029, which is 109^3 *Prime Curios<br /><br />Hundred, West Virginia was named for Henry Church and his wife, the first settlers who lived to be 109 and 106. Hundred is the only place in the United States with this name.<br /><br />109 is the product of two primes raised to their own power, \$ 109 = 2^2 * 3^3\$<br /><br />109 is remembered for being the number of the PT boat that future President Kennedy was on when it was hit in WWII.<br /><b><br /></b><b>110th Day of the Year</b><br />110 is the average of first fifty-three primes.<br /><br /> 110 is the side of the smallest square that can be tiled with distinct integer-sided squares (see image below). There are 3 distinct Simple Perfect Squared Squares with this property. Two 110's with 22 squares were discovered in 1978, one by Duijvestijn using computer search, the second by Willcocks, who transformed Duijvestijns 110 into a different second 110, and one more 110 with 23 squares was discovered in 1990 by Duijvestijn. It was Gambini who proved 110 is the minimal square. *http://www.archimedes-lab.org <br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-rYD_G288HeA/VxAQGzHxCtI/AAAAAAAAIK4/zsdMXDtnj7A3o7mkoWoJ4ZMa8p3XStC7wCLcB/s1600/110-Squared_square.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-rYD_G288HeA/VxAQGzHxCtI/AAAAAAAAIK4/zsdMXDtnj7A3o7mkoWoJ4ZMa8p3XStC7wCLcB/s400/110-Squared_square.jpg" /></a></div><br />110 = 5^2 + 6^2 + 7^2 (3 consecutive squares) = 11^2 - 11^1 (difference between powers of the same number)<br /><br />110 is the average of the first 53 prime numbers *Prime Curios, One wonders what percentage of the first n primes have an integer average?<br /><br />110 hertz is the standard frequency of the musical note A or La.<br /><br />110 is also known as "eleventy" according to the number naming system invented by J. R. R. Tolkien. <br /><br />110 is the smallest perfect number written in binary. But it is also the last Year date in decimal numbers that is a perfect number when read as a binary number.<br /><br />And 110 is the most commonly used impossible percentage, "Give 110%."<br /><br />110 is a pronic or oblong number, naturally representing the volume of some integer edged box. 110 cubic units is for a box of 2x5x11 units<br /><hr />111th Day of the year,<br /><br />If this day number, 111, in decimal digits is read as if it were binary, it would be 7, but it would also be the last day of the year that you can mistake for a binary number.<br /><br />111 would be the magic constant for the smallest magic square composed only of prime numbers if 1 were counted as a prime (and we often used to) It seems that Henry Ernest Dudeney may have been the first person to explore the use of primes to create a magic square. He gave the problem of constructing a prime magic square in The Weekly Dispatch, 22nd July and 5th August 1900. At that time, 1 was sometimes (often?) considered as a prime number. His magic square gives the lowest possible sum for a 3x3 using primes (assuming one is prime) <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-QV1VtQtid4Q/VxF25P5sdMI/AAAAAAAAILI/_stj6_lxR6Q2SnSihswz4_vdNYmHgbCnwCLcB/s1600/dudney%2Bprime%2Bsquare.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-QV1VtQtid4Q/VxF25P5sdMI/AAAAAAAAILI/_stj6_lxR6Q2SnSihswz4_vdNYmHgbCnwCLcB/s320/dudney%2Bprime%2Bsquare.jpg" /></a></div>The smallest magic square with true primes (not using one) has a magic constant of 177. Good luck. A six-by-six magic square using the numbers 1 through 36 also has a magic constant of 111. *Tanya Khovanova, Number Gossip<br /><br />Numbers like 111 that appear the same under 180<sup>o</sup> rotations are called strobograms. For numbers like the recent 109 which appears as a different number under rotation, but is still a number, I have created the term ambinumerals as an improvement on the commonly used "invertible".<br /><br /> If you concatenated three copies of 111 and then squared the result, you get (111,111,111)<sup>2</sup> = 12,345,678,987,654,321 *Cliff Pickover@pickover<br /><br /> Lagrange's theorem tells us that each positive integer can be written as a sum of squares with no more than four squares needed. Most numbers don't require the maximum four, but there are 58 year dates that can not be done with less than four. 111 and 112 are the smallest consecutive pair that both require the maximum. There is one other pair of consecutive year dates that also require four, seek them my children.<br /><br />111 is the sum of the non-prime numbers from 2 through 17. *Prime Curios<br /><br />If you started looking for primes using n(googol)+1, you want find one for a long time. Not until n = 111. The primes really get really spread out way up there...and still there are prime pairs as well. Bewitching mathematics!.<br /><br />111 is the smallest palindrome that has a prime digit sum.<br /><br />There are exactly 111 prime numbers that display on a digital clock. *Prime Curios<br /><br />111 is also a palindrome in base 6 (303) , and 111 in base 3, 5, 6, and 8 all convert to primes in base ten (7, 31, 43, 73) .<br /><br />The 6x6 magic square is sometimes called the Devil's square. It has a Magic Constant, or sum of each row or column of 111, but if yo add up all the numbers, you get 6 x 111 = 666, the so-called Number of the Beast.<br /><br />The British have no respect for their heroes. In cricket a score of 111 is called a Nelson, because the famous Admiral Nelson, by the end of his life, had one eye, one leg, and one arm. He might as well be Rodney Dangerfield.</div><div><br /></div><div>111 is odd, and like all odd numbers it is the difference of two consecutive squares that add up to the original number, so 56²-55² = 111, It is also the difference of two squares in a different way. All numbers that are equal to 3 Mod 6 (have a remainder of three when divided by six) are the difference of squares of numbers that differ by 3, and they are easy to find. If you divide 111 by 3 you get 37, so we need two numbers that add up to 37 and differ by 3..... easy algebra to find 20 and 17 giving us 20² - 17² = 111. <br /><br /><hr /><b>The 112th Day of the Year:</b><br />112 is a practical number (aka panarithmetic numbers), any smaller number can be formed with distinct divisors of 112. <i>Student's might explore the patterns of such numbers.</i><br /><br />112 is the side of the square that can be tiled with the the fewest number of distinct integer-sided squares, discovered by A. J. W. Duijvestijn in 1976 (see 110)<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-awjAOrZ61cU/VxJQotUa76I/AAAAAAAAILc/4Z9S_scg7aA7e5tx-gyaX0zYi-DHfapNwCLcB/s1600/21-Square.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-awjAOrZ61cU/VxJQotUa76I/AAAAAAAAILc/4Z9S_scg7aA7e5tx-gyaX0zYi-DHfapNwCLcB/s400/21-Square.gif" /></a></div><br />112 is the only 3-digit number such that its factorial raised to the sum of its digits and increased by one is prime. I.e., 112!<sup>(1+1+2)</sup>+1 is prime.<br /><br />112 = 11 + 13 + 17 + 19 + 23 + 29 (sum of consecutive primes) and = 1x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 (sum of oblong or pronic numbers) <br /><br />112 in binary looks like 111 followed by four zeros, 1110000, that makes it the sum of three consecutive powers of two, 2^4 + 2^5 + 2^6 = 112<br /><br />112 in base 3 is still only ones and zeros in a palindrome, 11011 The digit sum in base three and base ten are the same<br /><br />There are 112 pounds in a British long hundredweight.<br /><br />112 is a Harshad (Joy-giver) number, divisible by the sum of it's digits. And if you compute the Roman numeral letters for 112, CXII, by the alphabet code, A=1 etc then you get 3 + 24 +9 + 9=45, another Harshad number.</div><div><br /></div><div>In the Collatz (or 3n+1) sequence, the numbers 54 and 55 both take 112 steps to reach 1. No smaller number requires so many. The largest sequence of any year day is the 143 steps required for the number 327<br /><hr /><b>The 113th Day of the Year;</b><br />113 is prime, its reversal (311) is prime, and the number you get by any reordering of its digits is still prime. <i>Students might try to find other of these "absolute" or "permutable" primes. </i>There are two other three digit numbers, both year days, that have this same quality. There are also five 2 digit primes with this property, but that includes 11 which is sort of trivial.<br /><br /> Also the sum of the first 113 digits of e is prime. That was also true of yesterday's number, and tomorrow's. (I was just wondering to myself, what is the longest known string of consecutive n for which the first n digits of e are prime? And a similar question for pi? "Anyone...anyone??Bueller???)<br /><br /> \$ 113 \pi = 354.9999699.. \$ is almost an integer. No year day is closer, This was known to Chinese mathematicians by the end of the 5th century, "<span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">Zu Chongzhi (or Tsu Ch'ung Chi), along with his son Zu Gengzhi, stated in a mathematical text titled </span><i style="background-color: white; box-sizing: border-box; color: #212529; font-family: Roboto, sans-serif; font-size: 16px;">Zhui Shu</i><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"> (Method of Interpolation) that π is approximately three hundred fifty-five divided by 113." *Prime Curios</span><br /><span face=""roboto" , sans-serif" style="background-color: white; color: #212529;"> To remember this, I used to teach my students the jingle, "one one three three five five, divide in the middle, and put big over little." to remember that 355/113 is an approximation</span><span face=""roboto" , sans-serif" style="background-color: white; color: #212529;"> to pi for six digits, 3.1415929.... The error is less than (1/113)^2. </span><br /><br />There are 13 consecutive divisible integers (non-primes) between 113 and 127. How far until the next streak as long, or longer?<br /><br />113 was once the Atomic name of Element 113, <span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">ununtrium, which was later renamed Nihonium for the country of its discoverers, although a Russian team was also considered for the discovery. </span><br /><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;">If you raise the digits of 113 to any power from zero to four, the sum of the powers is a prime number, ex. 1^4 + 1^4 + 3^4 = 83</span><br /><span face=""roboto" , sans-serif" style="background-color: white; color: #212529; font-size: 16px;"><br /></span><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;">The sum of the digits of 113, 5, and the product of the digits , 3 are both primes, as is the sum of the squares of the digits, 11. </span></span><br /><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;"><br /></span></span><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;">113 is the largest known prime, P, for which there is no prime between P and P+sqrt(P) = (113-123) </span></span><br /><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;"><br /></span></span><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;">113 is a palindrome in base eight, 161. </span></span><br /><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;"><br /></span></span><br /><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;">113 is the smallest of five consecutive primes whose sum is prime. </span></span><br /><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;"><br /></span></span><span face=""roboto" , sans-serif" style="color: #212529;"><span style="background-color: white;">113 is the smallest integer that can not be represented in the Four-fours game under the standard rules. </span></span><br /><hr /><b>The 114th Day of the Year</b><br />114 begins a string of thirteen consecutive day numbers that are composite, marking a fourteen day prime gap (113-117). There is no string of more composite year day numbers. The next such string of composite day numbers will include Halloween. There is a prime gap of 114 between the six digit primes between 492113 and 492227. *Prime Curios<br /><br />The sum of the first 114 digits of e after the decimal point, is prime. This is the third consecutive day number with this property.<br /><br />The largest gap between two consecutive six digit primes is 114.<br /><br />114 is another of D R Kaprekar's Harshad (Joy-Giver) numbers, divisible by the sum of its digits. Remembering that the famous Taxicab number of Ramanujan and Hardy, is also a Harshad number makes it easy to factor, since 1 + 7 + 2 + 9 = 19 is a factor.<br /><br />114 is the sum of the first four hyperfactorials starting with zero, 0^0 + 1^1 + (2^2)(1^1) + (3^3)(2^2)(1^1) = 1+1+4+108 = 114.<br /><br />114 is a repdigit in base 7 (222) and a palindrome in base 5 (424)<br /><br />114 On September 6, 2019, Andrew Booker, University of Bristol, and Andrew Sutherland, Massachusetts Institute of Technology, found a sum of three cubes for \(42= (–80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 \). This leaves 114 as the lowest unsolved case . At the beginning of 2019, 33 was the lowest (positive integer) unsolved case, but Booker solved that one earlier in 2019. <br /><br /><hr /><b>The 115th Day of the Year</b><br />115 is the 26th "Lucky" number. <a href="https://en.wikipedia.org/wiki/Lucky_number" target="_blank">Lucky numbers</a> are produced by a sieve method created by Stan Ulam around 1955. The term was introduced in 1955 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve, "the sieve of Josephus Flavius" because of its similarity with the counting-out game in the Josephus problem. They are interesting explorations for both elementary and advanced students. Whether there are an infinite number of primes in the lucky numbers is still an open question. 115 (or 5! - 5) is the smallest composite number of the form p! - p, where p is prime.<br /><br /> \( \pi (115) = 30 \) occurs at the 115th decimal digit of pi. It is the smallest integer n, in which the number of primes less than n occurs at the nth decimal place of pi. Once more for the HS students, there are 30 prime numbers less than 115, and the 115th &116th decimal digits of pi are 3, 0, so the two digit value beginning at the 115th decimal place counts the number of primes less than 115. There is no smaller number for which this is true. You may want to find the next one. <br />Another way to get 30, is /( (1*1*5)^1 + (1*1*5)^2 /)<br /><br /> There are 115 ways (without including rotations and reflections) of placing 6 rooks on a standard chessboard so that they are not attacking<br /><br /><hr /><b>The 116th Day of the Year</b><br />116! +1 is prime. It is the 10th such year day, but only the second even number with the attribute.<br /> And: 116^2 + 1 is prime<br /><br />The number 1 appears 116 times in the first 1000 digits of pi. Thanks to *Math Year-Round @MathYearRound<br /><br />The sum of the first 116 digits of Pi is prime, the same is true for the first 117 and the first 118. This is the first occurrence of three consecutive numbers for which the sum of the digits of Pi to that point are prime. </div><div><br /></div><div>Impress your History teacher, the 100 Years war between France and England..... lasted 116 years. <br /><br />and Jiroemon Kimura died in 2013 in Japan. He was 116 years old. Two years later his record was broken by an even older Japanese citizen who died.<br /><br /> And for a bit of Americana, from a British web site called <i>*</i>isthatabignumber.com.. It's about Hyperion, a tree that is 116 meters tall.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-K37REc78zcI/W95MZm6YVGI/AAAAAAAAJOI/BT1xCDWNeiI1bxn67CDA-XtXat5ng7ujQCLcBGAs/s1600/tallest%2Btree.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="357" data-original-width="310" height="320" src="https://1.bp.blogspot.com/-K37REc78zcI/W95MZm6YVGI/AAAAAAAAJOI/BT1xCDWNeiI1bxn67CDA-XtXat5ng7ujQCLcBGAs/s320/tallest%2Btree.jpg" width="277" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><b>------------------------------------------------------------------------------------------------------------</b></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><b>The 117th Day of the Year</b></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;">117 is the smallest possible length of the longest side of a </span><a href="https://en.wikipedia.org/wiki/Heronian_tetrahedron" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; text-decoration-line: none;" title="Heronian tetrahedron">Heronian tetrahedron</a><span face="sans-serif" style="background-color: white; color: #222222;"> (one whose sides, face areas, and Volume are all integers) *Prime Curios</span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;">An Euler brick, named after Leonhard Euler, is a cuboid whose edge and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.The smallest Euler brick, discovered by Paul Halcke in 1719, has edges(a,b,c) = (44, 117, 240) and face diagonals 125, 244, and 267. </span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><span style="color: black;">The sum of the first 116 digits of Pi is prime, the same is true for the first 117 and the first 118. This is the first occurrence of three consecutive numbers for which the sum of the digits of Pi to that point are prime. </span></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><span style="color: black;"><br /></span></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;">117^2 is prime when two is added, or subtracted. The larger 13691, is the smaller of a pair of twin primes, but 13687 is not. </span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;">117 can be written as the difference of two squares, or of two cubes. \$11^2 - 2^2 = 5^3 - 2^3 = 117 \$ </span><span face="sans-serif" style="background-color: white; color: #222222;">(Can you find another number which can be expressed as both the difference of squared primes and cubed primes?)</span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;">117 is a repdigit in base 12(99) and a palindrome in base 6(313).</span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span face="sans-serif" style="background-color: white; color: #222222;">117 is the sum of three consecutive powers of three, 3^2 + 3^3 + 3^4 = 117</span></div><hr /><b>The 118th Math Day of the Year</b><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">118 is the smallest n such that the range n, n + 1, ... 4n/3 contains at least one prime from each of these forms: 4x + 1, 4x - 1, 6x + 1 and 6x - 1.</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">118 is the smallest even number not differing by one or a prime number from one of its prime neighbors.</span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;">*Prime Curios</span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222;"><br /></span><span style="background-color: white;"><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="color: #222222;">There are four unique partitions of 118 into three integers that all have the same product. No smaller example exists. </span></span><span face="sans-serif" style="background-color: white; color: #222222;">14 × 50 × 54 = 15 × 40 × 63 = 18 × 30 × 70 = 21 × 25 × 72 = 37800. </span><br /><br />In the spelling, ONE HUNDRED EIGHTEEN, one hundred uses 11 character spaces, Eighteen uses 8 , concatenated, we have 118.<br /><hr /><b>The 119th Math Day of the Year</b><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; text-align: justify;">the largest amount of US money one can have in coins without being able to make change for a dollar is 119 cents. *Tanya Khovanova, Number Gossip (3 quarters, 4 dimes, 4 pennies)</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; text-align: justify;">119 is the product of the first two primes ending with 7</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify;" />119 is a palindrome in base 2, (1110111), and in base 16, (77)<br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; text-align: justify;">119 is the order of the largest cyclic subgroups of the Monster group.</span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; text-align: justify;"><br /></span><span style="background-color: white; text-align: justify;"><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="color: #222222;">119 is the sum of five consecutive primes, beginning with its largest prime factor. It is the 7th of 18 year days which are the sums of five consecutive primes. Do any of of these other sequences include one of the prime factors?</span></span><br /><br />119 is the smallest composite number, and the only year date, that is one less than a factorial. The next will be 40319 = 8! - 1. (students might examine the sequence of n! + 1 for patterns)<br /><br />119 is a Perrin Number, A Fibonacci like sequence that begins with 3, 0, 2 and then each new value is the sum of the two digits before the last known, so it starts 3, 0, 2, 3, 2, 5, 5, 7, ... The name is for French mathematician Francois Perrin who wrote about it in 1899,<br /><hr /><b>The 120th Math Day of the Year</b><br />120 is a Harshad (Joy-giver) number, divisible by the sum of its digits, 120 is the smallest three-digit Harshad number for which the quotient of that division is also a Harshad Number (and that quotient is again a Harshad number. ) Wondering if other Harshad numbers for which the quotient is another, then chain down to a division by one or the number one itself.. 162 seems to follow a similar path, but ends in 1<br /><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">The 120th day of the year; All primes (except 2 and 3) are of form 6*n +/- 1. Note that 120 = 6*20 is the smallest multiple of six such that neither 6n+1 or 6n-1 is prime. *Prime Curios Can you find the next. I do find it interesting that both 119 and 121 have exactly two factors, and both end in the same digit (7x17) and 11x11)</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">120 = 3¹ + 3² + 3³ + 3⁴</span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">Had to add this one,120 is the smallest number to appear 6 times in Pascal's triangle. *What's Special About This Number</span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">(There are only three days of the year that appear in the arithmetic triangle more than five times. What are the other two?)</span><br /><br />120 is the sum of four consecutive primes, four consecutive powers of two, and four consecutive powers of thee. (desperately seeking a fourth sum of four things for the symmetry of this)<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-kyrM7ft75bw/XnZnSKjcpOI/AAAAAAAALl4/MYv05GP22eMrR8HGGv_VGAdd9MfGW2sQwCLcBGAsYHQ/s1600/120%2Bcell%2B4%2Bd%2Bregular%2Bsolid.png" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="246" data-original-width="246" src="https://1.bp.blogspot.com/-kyrM7ft75bw/XnZnSKjcpOI/AAAAAAAALl4/MYv05GP22eMrR8HGGv_VGAdd9MfGW2sQwCLcBGAsYHQ/s1600/120%2Bcell%2B4%2Bd%2Bregular%2Bsolid.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Wikipedia</td></tr></tbody></table>The hecitonicosachron is a four-dimensional regular convex solid, with 120 dodecahedral 3-d sections. There are six of these regular convex structures in 4 space, compared to the Five Platonic solids in three space. Beyond that, all known dimensions have less of them.<br /><br style="background-color: white;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">6 and 28 are prefect numbers because the sum of their proper divisors is equal to the number. 120 is the only year date that is a multi-perfect number. The sum of its proper divisors is 2 * 120. (known since antiquity, the second smallest , discovered by Fermat in 1636, is 672. It has been conjectured that there are only six of these. The fate of these "tri-perfect"(so called because the sum of all the divisors, including the number itself, is three times the original number) numbers is related to the search for and odd perfect number, if n is perfect, then 2n is a tri-perfect number. </span><br /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;" /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">120 is the largest number of spheres that can contract a central sphere in eight dimensions. Beyond the fourth dimension, this "kissing number" is only known for the eighth and 24th dimensions.</span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">If you list the divisors of 120, add their reciprocals, you get a prime integer. There is no smaller number that has an odd prime sum. Students can search for the smaller number which has a sum of the divisors reciprocals of an even prime. </span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">120 is 5!, and it is one less than a perfect square, thus leading to one solution of Brocard's problem, find m,n so that m!+1 = n^2. Since 120 is a factorial itself, you can express its factorial as the product of two other factorials... 120! = 120*119! ,or in general (n!)! = (n!)(n!-1)!.</span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br />Speaking of 5!=120, if you make a list of the fifth powers of the consecutive integers, and then make a list below of their differences, and another list below of those differences, you get..... oh , watch:</span></div><div><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><div><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">1.........32........243.........1024........... 3125.........7776........16807</span></div><div><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">.....31..........211.........781..........2101........4651...........9031</span></div><div><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">............180.........570.........1320.........2550...... 4380</span></div><div><span style="color: #222222;"><span style="background-color: white; font-size: 13.2px;">....................390 ........750............1230.......1830</span></span></div><div><span style="color: #222222;"><span style="background-color: white; font-size: 13.2px;">..............................360.........480............600</span></span></div><div><span style="color: #222222;"><span style="background-color: white; font-size: 13.2px;">......................................120,,,,,,,,,,,120............120.... </span></span></div><div><span style="color: #222222;"><span style="background-color: white; font-size: 13.2px;">And in general, the nth differences of the nth powers of the integers is n! *Conway and Guy, The Book of Numbers</span></span></div><div><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;">120 is the sum of a twin prime pair, (59+61) </span><br /><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span><span style="background-color: white;"><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">120 is a triangular number, the sum of the first 15 positive integers, and also a tetrahedral number, the sum of the first eight triangular numbers. It is also the only triangular number that can be expressed as the product of three, four, or five consecutive integers, 4 x 5 x 6 = 2 x 3 x 4 x 5 = </span></span></span><span style="background-color: white; color: #222222; font-size: 13.2px;"> 1 x 2 x 3 x 4 x 5. </span><br /><span style="background-color: white;"><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="color: #222222;"><span style="font-size: 13.2px;"><br /></span></span></span><span style="background-color: white;"><span face=""arial" , "tahoma" , "helvetica" , "freesans" , sans-serif" style="color: #222222;"><span style="font-size: 13.2px;">History lesson for young people, The Kodak Brownie Number 2, produced from 1901 to 1935, was the first camera to use 120 film. The three models (cardboard case to aluminum case) complete with view finder and handle, cost less than $2.75 US. Many are still in use by professional photographers. Imagine using your cell-phone camer when it's 75 years old. </span></span></span></div></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-41974794488506811932020-02-18T11:28:00.044-08:002021-04-02T07:33:33.242-07:00Number Facts for Every Year Day (61-90) from On This Day in Math<b>The 61st Day of the Year:</b>The 61st Fibonacci number (2,504,730,781,961) is the smallest Fibonacci number which contains all the digits from 0 to 9 *Tanya Khovanova, Number Gossip (are there others that contain only the first 2, 3 .. 9 digits? <i>ie 21 has 1,2 but 121393 has 1,2,3 but also a 9. Is there any that contain ONLY 1,2,3 or 1,2,3,4 etc</i>?)<br /><br />In 1657, Fermat challenged the mathematicians of Europe and England, "We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul (Fermat's region) will." Among the challenges was this 500-year-old example from Bhaskara II: x2 - 61y2 = 1 (x, y > 0). *Prime Curios<br /><br />Among all the primes less than 10^9, the final two digits most common is 61.<br /><br />As a prime number of the form 4n+1, 61 can be written as the sum of two squares in only one way, 5^2+6^2 .<br /><br />61# (read 61 primorial, the product of 61 and all lesser primes) is the smallest Primorial that is Pandigital (Contains all the numerals 0-9). Guess you could say it is the most Petite Primorial Pandigital.<br /><br />If you multiply 61 by its digit reversal, 16, and then add one, you get a prime. The smallest multidigit prime for which this is true.<br /><br />The smallest prime that can be written as the sum of a prime number of primes to prime powers in a prime number of ways: \(2^2 + 2^3 + 7^2\), \(2^2 + 5^2 + 2^5\), and \(3^2 + 5^2 + 3^3\).*Prime Curios<br /><br />If you divide 61! by 16! and add one, you get a prime number which has a prime number of digits and a prime sum of its digits. If instead you multiply them, and subtract 1, you get another prime with a prime sum of digits. *Prime Curios<br /><br />And if you start searching pi at the 61st digit after the decimal point, you will find a string of all ten distinct numerals.<br /><br />The reciprocal of 61 has a repeating decimal of length 60. 1/61 = 0.016393442622950819672131147540983606557377049180327868852459... Primes, p, with reciprocals of length p-1; have the unique property that the first n/2 digits have a 9's compliment in the same position of the next n/2 digits (for a simple example, 1/7 = .142857.. and 1+8 = 4+5 = 2+7 = 9 ) Joshua Zucker corrected me for calling this "unique", as there are other primes with periods less than n-1 that share this property of compliments, 1/13 for example. By the way, that reciprocal is the smallest prime with the same equal numbers of all ten numerals.<br /><br />If you take the five consecutive primes from 61 to 79, and arrange them in a 3x3 square matrix, then the rows and columns sum to three more consecutive primes ( 199, 211, 223) *Prime Curios<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-EZtKOJYnKQo/XkGg-NQb7ZI/AAAAAAAALMI/Souu3laGI10ZOoODqrF1R17--xFlhiJxgCLcBGAsYHQ/s1600/consecutive%2Bprimes%2Bmatrix.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="108" data-original-width="120" height="285" src="https://1.bp.blogspot.com/-EZtKOJYnKQo/XkGg-NQb7ZI/AAAAAAAALMI/Souu3laGI10ZOoODqrF1R17--xFlhiJxgCLcBGAsYHQ/s320/consecutive%2Bprimes%2Bmatrix.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Prime Curios </td></tr></tbody></table><br /><br /><br />If A=1,... Z=26, then the word PRIME is prime. (P+R+I+M+E --> 16+18+9+13+5 =61 61 is the exponent of the 9th Mersenne prime.(2<sup>61</sup> − 1 = 2,305,843,009,213,693,951) (But amazingly, somehow Mersenne omitted this exponent in his list of conjectured primes.<br /><br />Sixty-one has no repeat letters, and if you spell out any larger prime in English, you will never find another with no repeated letters.<br /><br />61 is also a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61.. (Keith numbers were introduced by Mike Keith in 1987 who called them repfigit number, short for repetitive Fibonacci-like digit). They are computationally very challenging to find, with only about 100 known.)<br />AND the 61st Fibonacci number, is the first Pandigital Fibonacci Number, 2504730781961.<br />AND no odd Fibonacci number is divisible by 61.<br /><br />On June 30, of 2012, a Leap Second was added to the clock, and created a minute with 61 seconds.<br /><hr /><b>The 62nd Day of the Year:</b><br />62 is the smallest number that can be written as the sum of 3 distinct squares in 2 ways. (Students might try to find the smallest number that can be written as the sum of 2 distinct squares in 2 ways)<br /><br />In base 10, 62 is also the only number whose cube (238328) consists of 3 digits each occurring 2 times<br /><br />The prime factorization of 62 is 2*31. There are only two numbers whose prime factorization uses only the first three counting numbers once each in its digits. The other is its digit reversal, 26 = 2*13<br /><br /> If you average the first n digits of pi after the decimal point, sometimes the average is an integer (for example, 1+4+1 = 6 and 6/3 = 2 so the first three digits work). 62 digits is the highest known number of digits that work. There is actually a good reason for this, the digits of pi are essentially random, and so they would average 4.5 in the long run. While small numbers may vary more from this value, eventually the values will approach 4.5 within a boundary of less than 1/2, so no integers.<br /><br />62 is the largest known even number that cannot be expressed as the sum of two odd semiprimes. The sum of the prime factors of this semiprime, 62, gives the largest known number that cannot be expressed as the sum of any two semiprimes. <br /><br /><br />The digits 62 occur at the 61st & 62nd digits of phi, φ; AND The 61st & 62nd digits of e.<br /><br />Jim Wilder pointed out a strange amicable relation ship involving 62, 69 and their squares:<br />\$ 62^2 + 05^2 = 3869\$ but then<br />\$ 38^2 + 69^2 - 6205\$ <br /><br /> and 62 is supposedly the age at which Aristotle died.<br /><br /><a href="https://4.bp.blogspot.com/-F_cU7lijMAo/V-m7S9Td8fI/AAAAAAAAIek/zbAjGIxL0Wsk7sl9L-L93ONBzwlg4jylgCLcB/s1600/possum%2Btrot.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="150" src="https://4.bp.blogspot.com/-F_cU7lijMAo/V-m7S9Td8fI/AAAAAAAAIek/zbAjGIxL0Wsk7sl9L-L93ONBzwlg4jylgCLcB/s200/possum%2Btrot.jpg" width="200" /></a>If you start at the beginning of Pi3.14... the 62nd digit begins a string of all ten distinct numerals.<br /><br />62 in base ten is a repdigit in bases 5 (222<sub>5</sub>) <br /><br />And if you ever want to visit Possum Trot, Ky, just get on US 62, and watch for the sign... but don't blink.<br /><hr /><b>The 63rd Day of the Year:</b> <br /><a href="https://3.bp.blogspot.com/-Fl1g_Oz-bTE/VtCw2yCdq0I/AAAAAAAAH6Q/Z2zKW4S5zkU/s1600/100_views_edo_063.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://3.bp.blogspot.com/-Fl1g_Oz-bTE/VtCw2yCdq0I/AAAAAAAAH6Q/Z2zKW4S5zkU/s200/100_views_edo_063.jpg" width="131" /></a> in Roman Numerals 63 is LXIII. If you represent each of these letters by its number in the English alphabet you get 12+24+9+9+9=63. (<i>There is one more number that has this quality.</i>)<br /><br /> At right, in honor of my many students from Misawa, Aomorishi, Japan, is Print 63 of Utagawa Hiroshige's 100 views of Edo (Koi No Bori)<br /><br /> \( \phi(63) = 36\) The number of positive integers which are less than 63 and relatively prime to it.<br /><br /> 63 can be expressed as powers of its digits, \( 6^2 + 3^3 = 63\) <br /><br />63 is the Fourth Woodall Number. Numbers of the form n*2<sup>n</sup>-1. 63 = 4*2,sup>4</sup> -1 Woodall Numbers were used in the study of testing prime numbers. There is only one more Woodall Number that is a Day of the Year.<br /><br />The Five Factorials Game, 2! * 5! / 3! + 4! - 1! = 63<br /><br />63 is the smallest whole number that can be divided by any number from 1 to 9 without repeating decimals. (What's Next?)<br /><hr /><b>The 64th Day of the Year:</b><br />64 is the smallest power of two with no prime neighbor. (What is next value of 2<sup>n</sup> with no prime neighbor?)<br /><br />64 is also the first whole number that is both a perfect square and a perfect cube.<br /><br />The sixth Fermat Prime, 2^64 +1 was factored by F. Landry in 1880 as the product of 274177 and 67280421310721. The next Fermat Prime would not be factored until 1970.<br /><br /> There were 64 disks in Eduard Lucas' myth about the Towers of Hanoi. 64 is also the number of hexagrams in the I Ching, and the number of sexual positions in the Kama Sutra. (I draw no conclusions about that information)<br /><br />64 can be expressed as the sum of primes using the first four natural numbers once each, 41 + 23 = 64.<div>It can also be done to its reversal 46 = 41 + 3 + 2</div><div><br /> There are 64 ordered permutations of nonempty subsets of {1,..., 4}: Eighteenth- and nineteenth-century combinatorialists call this the number of (nonnull) "variations" of 4 distinct objects. <br />64 is the smallest number with exactly seven divisors. <br /><br />64 is the smallest square that creates two primes if concatenated with its previous and next squares, i.e., 6449, 6481.*Prime Curios<br /><br />64 is also the smallest square without a prime neighbor.<br /><br />64 is a superperfect number—a number such that σ(σ(n)) = 2n. The sum of the divisors (including itself) of 64 is 127, and the sum of the divisors of 127, 1 and 127, add up to 128= 2*64. It is the last Year Day that is Super-Perfect. <br /><br />And I was told that 64 is the maximum number of strokes used in a Kanji character.</div><div><br /></div><div>Most mathematicians know the story of 1729, the taxicab number which Ramanujan recognized as a cube that was one more than the sum of two cubes, or the smallest number that could be expressed as the sum of two cubes in two different ways. But not many know that 103 is part of the second such \(64^3 + 94^3 = 103^3 + 1^3 \) <br /><hr /><b>The 65th Day of the Year:</b><br />65 is the smallest hypotenuse of two different primitive Pythagorean triangles (and of two other triangles that are not primitive) with all integral sides. (<i>Don't just sit there, find them!</i>) 65 is a 4n+1 semiprime that is factorable into a pair of 4n+1 primes.</div><div>John Golden @mathhombre not found them all, he made a beautiful graphic of them. </div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-lVrjLWalAaM/YETRyn_XnEI/AAAAAAAANZY/qLDYo_1oUTsrh_tPxHyb1lWYDQYHwS6iwCLcBGAsYHQ/s900/65%2Bas%2Bpythag%2Bhypot.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="670" data-original-width="900" src="https://1.bp.blogspot.com/-lVrjLWalAaM/YETRyn_XnEI/AAAAAAAANZY/qLDYo_1oUTsrh_tPxHyb1lWYDQYHwS6iwCLcBGAsYHQ/s320/65%2Bas%2Bpythag%2Bhypot.jpg" width="320" /></a></div><br /><div><br /><br />Primes of the form 4n+1 can always be formed as the sum of two squares. Primes can only be expressed as such a sum in one way. Since 65 is a semiprime number with two 4n+1 Fermat Prime factors, it can be expressed as the sum of two squares in two ways. 8^2 + 1^2 = 65 = 7^2 + 4^2<br /><br />And \( 65 = 1^5 + 2^4 + 3^3 + 4^2 + 5^1 \) *jim wilder @wilderlab<br /><br />OR, \(65= 0^2 + 1^4 + 2^5 + 3^3 + 4^1 + 5^0 \) *@Expert_says <br /><br /><a href="http://4.bp.blogspot.com/-kodLmRrphv0/VPB88YFU0vI/AAAAAAAAGZo/ovDiamsGGR0/s1600/5x5magic%2Bsquare.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://4.bp.blogspot.com/-kodLmRrphv0/VPB88YFU0vI/AAAAAAAAGZo/ovDiamsGGR0/s320/5x5magic%2Bsquare.png" /></a>65 is the constant of a 5x5 normal magic square. A magic square with the integers 1 through 25 has a sum of 65 in each row, column, and major diagonal.<br /><br /> Euler found 65 integers, which he called "numeri idonei," that could be used to prove the primality of certain numbers.[idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as \(x^2 ± Dy^2\) (where x<sup>2</sup> is relatively prime to Dy<sup>2</sup>) is a prime, prime power, twice one of these, or a power of 2. In particular, a number that has two distinct representations as a sum of two squares (such as 65) is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.]<br /><br />The Five Factorials Game, 2! * 5! / 3! + 4! + 1! = 65<br /><br />65 is the difference of fourth powers of two consecutive primes. And a note about fourth powers of primes. For any prime greater than five, the last digits of a p^4 either ends in an odd digit followed by six, or an even digit follwed by one.<br /><br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">65 = 1</span><sup style="background-color: white; color: #222222; font-family: sans-serif; font-size: 11.2px; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> + 2</span><sup style="background-color: white; color: #222222; font-family: sans-serif; font-size: 11.2px; line-height: 1;">4</sup><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> + 3</span><sup style="background-color: white; color: #222222; font-family: sans-serif; font-size: 11.2px; line-height: 1;">3</sup><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> + 4</span><sup style="background-color: white; color: #222222; font-family: sans-serif; font-size: 11.2px; line-height: 1;">2</sup><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> + 5</span><sup style="background-color: white; color: #222222; font-family: sans-serif; font-size: 11.2px; line-height: 1;">1</sup><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">.</span><br /><hr /><b>The 66th Day of the Year:</b><br /><a href="http://2.bp.blogspot.com/-iu9htYqAVz0/VPEt_gv9skI/AAAAAAAAGaM/fnJZuYPgfIA/s1600/route%2B66.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-iu9htYqAVz0/VPEt_gv9skI/AAAAAAAAGaM/fnJZuYPgfIA/s1600/route%2B66.jpg" width="193" /></a> there are 66 different 8-polyiamonds (A generalization of the polyominoes using a collection of equal-sized equilateral triangles (instead of squares) arranged with coincident sides.)<br /><br /> Route 66, known as the Main Street of America was dubbed the "Mother Road" by novelist John Steinbeck, The strobogrammatic partner of 66 is 99, and the Former US 99 was dubbed the "Mother Road" of California during the dust bowl era.<br /><br />66 is the smallest strobogrammatic number that shares a strobogrammatic factor with its partner, 99.<br /><br />66 is the smallest number where the sum of its divisors is a perfect square. \(1 + 2 + 3 + 6 + 11 + 22 + 33 + 66 = 12^2 = 144\) There are only three year dates (to my knowledge) for which the sum of the divisors is a square. They all three occur in this MathFacts page, 66, 70, and 81. 66 and 70 even have the same total. <br /><br /> If you wrote out all the numbers on a 12 hour clock, (HrMin, so 6:03 would be 603, etc.), there would be 66 of them that are prime.<br /><br /><br />The smallest number that can be expressed as a sum of two two-digit primes, each ending with the digit three, in two different ways (66 = 13 + 53 = 23 + 43) and the smallest number that can be expressed as the sum of two primes in six different ways.<br /><br />66 is the largest day number of the year which does not have a letter "e" in its English spelling. Sixty-six is the 19th such numbers in the year, but the next number without an "e" is 2000.<br /><br /><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">66 however is not the sum of two squares. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">To check if an arbitrary number is the sum of two squares, factor it. If any factor p^a + 1 is divisible by four, then it is not a sum of two positive squares. For 66 = 2 x 3 x 11, the factor of three (or 11) is the killer since one more than either is divisible by four. </span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222; font-size: 13.2px;"><br /></span></div><br /><br />6 is a triangular number, and 66 is as well, and, Oh Heck Yeah, 666 is too.... And they are all palindromes as well, but 66 is the eleventh triangular number, the only one of these with a palindrome index. (note 6666 is not a triangular number, but 66066 is and it is the 363 triangular number, so its index is a palindrome as well. Only six triangular numbers less than \(10^10\) are palindromes with a palindrome index.)</div><div><br /></div><div>And like all triangular numbers, 8 x T + 1 is a square number. For 6, 66, and 666 the squares produced are 7^2, 23^2, and 73^2.<br /><br />Sixty Six is an unincorporated community in Orangeburg County, South Carolina, United States. Sixty Six is located along U.S. Route 21, north of Branchville. <br /><br /><div style="text-indent: -16px;"> 66 is a semi-perfect number, since some subset of it's proper divisors add up to 66.(11+22+33=66 there is another way to do it also )<br /><br /> And just to conjure up one more variant on the number of the beast, 666, there are exactly 66 six-digit primes with distinct non-prime digits. </div><hr /><b>The 67th Day of the Year: </b><br />67 is the largest prime which is not the sum of distinct squares.<br /><br />Mersenne thought 2^67 - 1 was prime.; In 1867 Lucas proved that it was not prime, but could not find the factors. It was not until October 31, 1903 that Frank Nelson Cole found the factors. ;During Cole's so-called "lecture", he approached the chalkboard and in complete silence proceeded to calculate the value of M67 with the result being 147,573,952,589,676,412,927. Cole then moved to the other side of the board and wrote 193,707,721 × 761,838,257,287, and worked through the tedious calculations by hand. Upon completing the multiplication and demonstrating that the result equaled M<sub>67</sub> Cole returned to his seat, not having uttered a word during the hour-long presentation. His audience greeted the presentation with a standing ovation. Cole later admitted that finding the factors had taken "three years of Sundays." *Wik<br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><br /></span>It is the 19th prime number and the sum of five consecutive primes ending in 19 (7 + 11 + 13 + 17 + 19)<br /><br />The maximum number of internal pieces possible if a circle is cut with eleven lines. These are sometimes called "lazy caterer's numbers." \( 67 = \binom{11}{0} + \binom {11} {1} + \binom {11}{2} \)<br /><br /> 67 is also the smallest prime which contains all ten digits when raised to the tenth power. *Prime Curios<br />67^10 =1822837804551761449<br /><br />67 in Roman Numerals if LXVII. If you evaluate that using the a=1, b=2 method, the sum LXVII = 76, the reversal of the digits of 67.<br /><br />67 is the largest known prime for which 2^p does not contain any zeros.<br /><br />and Jim Wilder @wilderlab sent 67 = 2<sup>6</sup> + 2<sup>1</sup>+ 2<sup>0</sup> = 26 + 21 + 20 = 67<br /><br /><br />and @expert_says added several more. Students might be shown one relationship, and encouraged to discover others:<br />67=1² + 1² + 1⁷ + 2⁶ =12+12+17+26 <br />67=1² + 1³ + 1⁶ + 2⁶ =12+13+16+26<br />67=1² + 1⁴ + 1⁵ + 2⁶= 12+14+15+26<br />67=1³ + 1³ + 1⁵ + 2⁶=13+13+15+26<br />67=1³ + 1⁴ + 1⁴ + 2⁶=13+14+14+26<br /><br /><br />And one <a href="http://pballew.blogspot.com/2008/05/taking-things-to-new-and-some-old.html" target="_blank">smoot</a> is equal to 67 inches. <b><br /></b><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tXmL7IvD304/XkHV2ZVyAZI/AAAAAAAALMU/h1a-feJnZ3I1QcHy_XgitD8ZrqBhonYmACLcBGAsYHQ/s1600/foucault%2Bpantheon.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="341" data-original-width="400" height="170" src="https://1.bp.blogspot.com/-tXmL7IvD304/XkHV2ZVyAZI/AAAAAAAALMU/h1a-feJnZ3I1QcHy_XgitD8ZrqBhonYmACLcBGAsYHQ/s200/foucault%2Bpantheon.jpg" width="200" /></a></div>Foucault's Famous Pendulum had a length of 67 meters. You can see it now in the Beautiful Museum of Arts and Crafts.<br /><br />67 is palindromic in the consecutive bases 5 (232<sub>5</sub>) and 6 (151<sub>6</sub>).<br /><br /><br /><hr /><b>The 68th Day of the Year:</b><br />if you searched through pi for all the two digit numbers, the last one you would find is 68. The string 68 begins at position 605 counting from the first digit after the decimal point. (What is the last single digit numeral to appear? One might wonder how far out the string would you have to go to find all possible three digit numbers? )<br /><br />68 is the largest known number to be the sum of two primes in exactly two different ways:<br /><br />68=7+61=31+37. <span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> </span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">All higher even numbers that have been checked are the sum of three or more pairs of primes</span><br /><br />68 is a stobogrammatic number, rotated it is 89. Some consider only invertible numbers (rotated they form the same value, like 181) as strobograms. HT to Paul O'Malley By the way, it's the smallest composite number with a prime strobogrammatic partner.<br /><br />There are exactly 68 ten digit binary numbers in which each digit is the same as one of it's adjacent digits.<br /><br />68 is the smallest composite number that can be read as a prime number when it is rotated 180<sup>o</sup> HT Jim Wilder @wilderlab.<br /><br />And a historical oddity, in 46 BCE, as a result of Julius Caesar's Calendar adjustment, there were 68 days inserted between November and December. <b><br /></b> 68 is also a Happy number since 68 → 6^2 + 8^2 = 100 → 1^2 + 0^2 + 0^2 = 1 <br /><br />\( 2^68\) is the smallest power of two that is pandigital, with all ten decimal digits. It is a number 21 digits long. *@Fermat's Library<br /><hr /><b>The 69th Day of the Year:</b><br />the square and the cube of 69 together contain all ten numerals. 69<sup>2</sup> = 4761, 69<sup>3</sup> =328509<br /><br />10<sup>69</sup>+69 is prime and; 100<sup>69</sup>-69 is prime<br /><br />On Many scientific calculators, 69! is the largest factorial that can be calculated, with an overflow error for larger numbers. 69! is appx 1.711 (10<sup>98</sup>)<br /><br /> Don S. McDonald @McDONewt pointed out that \( \binom{69}{5}\) = 11238513, 7 Fibonacci #'s <i>almost</i> in order.<br /><br />69 is a strobogrammatic pair with itself. It is the smallest such number with distinct numbers. Reversing the order of its digits gives 96, another strobogrammatic self-pair. And for a bonus, the sum of the divisors of 69 is 96. <br /><br />The first squared square to be found was a square filled with 69 different smaller squares. ( electrical network theory was used to make the discovery, previously most mathematicians felt that were not likely to be any squared squares <i>see Jan 21</i>).. The first squared square was published in 1938 by Roland Sprague who found a solution using several copies of various squared rectangles and produced a squared square with 55 squares, and side lengths of 4205<br />No squared square can be made with less than 21 squares(shown below)<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-C3VAVs7NAlY/VPUR5TqnxQI/AAAAAAAAGcU/ETPamceE6Vg/s1600/squared%2Bsquare.png" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://3.bp.blogspot.com/-C3VAVs7NAlY/VPUR5TqnxQI/AAAAAAAAGcU/ETPamceE6Vg/s320/squared%2Bsquare.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Lowest-order perfect squared square *Wik</td></tr></tbody></table><b>Jim Wilder</b> sent me these facts involving, among others, the number 69.<br /><div>\$ 3869 = 62^2 + 05^2 \$<BR>\$ 6205 = 38^2 + 69^2 \$ </div><div>I quickly seized the moment to coin the term for these as Amicable Sums of Squares (just don't abbreviate it.)<br /><hr /><b>The 70th Day of the Year: </b><br />70 is the smallest "Weird" number. In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.</div><div><br /></div><div>All the primes in the 70's, are emirps, primes that are still prime when you reverse the order of the digits, 71----17 etc.<br /><br />2<sup>70</sup> = 1180591620717411303424. The sum of the digits is 70, and if you reverse the order, 424303114717026195081, it is a prime #.<br /><br /> \( 1^2 + 2^2 + 3^2 + \cdots + 24^2 = 70^2 \) Such numbers (sums of first n squares) are called Square Pyramidal Numbers (This one actually has a relationship to the Leech Lattice and String Theory)<br /><br />Several languages, especially ones with vigesimal number systems, do not have a specific word for 70: for example, French soixante-dix "sixty-ten"; Danish halvfjerds, short for halvfjerdsindstyve "three and a half score". (For French, this is true only in France; other French-speaking regions such as Belgium, Switzerland, Aosta Valley and Jersey use septante.) *Wik </div><div><br /></div><div>70 is the second smallest number where the sum of its divisors is a perfect square. \(1 + 2 + 5 + 7 + 10 + 14 + 35 + 70= 12^2 = 144\) There are only three year dates (to my knowledge) for which the sum of the divisors is a square. They all three occur in this MathFacts page, 66, 70, and 81. 66 and 70 even have the same total. <br /><hr /><b>The 71st Day of the Year:</b><br />71<sup>2</sup>=5041 = 7! +1! *<a href="http://primes.utm.edu/" target="_blank">Prime Curios</a> 4! +1, and 5!+1 are also squares but not the factorial of the digits. Whether there is a larger value of n for which n! + 1 is a perfect square is still an open question, called the Brocard problem after Henri Brocard who asked it in 1876. It has been proven that no other numbers exist less than 10<sup>9</sup>. <a href="http://amzn.to/1U4MdXf">*Professor Stewart's Incredible Numbers</a> Cliff Pickover, wrote that 71 is the largest known prime, p, such that p<sup>2</sup> is the sum of distinct factorials.<br /><br />A Cute way of writing the above, \(\sqrt{7! + 1} = 71 \)<br /><br />71 is the first two decimal digits of the expression of e=2.718281828459045...<br /><br /><br />71 is the largest number that occurs as a prime factor of the order of a sporadic group.*Wik<br /><br />71 is the first of three consecutive primes that are all still primes when their digits are reversed. (is there another such occurrence?) All the primes in the 70's, are emirps, primes that are still prime when you reverse the order of the digits, 71----17 etc.<br /><br /><i>and too good to leave out</i>, 71 is the only two-digit number n such that (n<sup>n</sup>-n!)/n is prime. *Tanya Khovanova, Number Gossip (<i>Be the first on your block to find a three digit example.</i>)<br /><br />In 1935, Erdős and Szekeres proved that 71 points (no three on a single line) are required to guarantee there are six that form a convex hexagon, although 17 points are thought to be sufficient. (In 1998, the upper bound was reduced to 37.) *Prime Curios<br /><br />71<sup>3</sup>=357,911 where the digits are the odd numbers 3 to 11 in order * @Mario_Livio<br /><br /> 71<sup>3</sup> is also the only cube of a 2-digit number that ends in 11. There is only one 1digit cubed that ends in 1, and only one three digit cubed that ends in 111(<i>Don't just sit there children, go find them</i>!). Could there be a four digit cube that ends in 1111<br /><br />The sum of all the prime numbers up to, and including 71 is not prime, but it is divisible by 71. (This works for 5 as well, and for 369119, and that's the only ones we know about.) 71 will also divide the sum of all the primes smaller than 71. It is the smallest such prime to be a proper divisor of the sum of all smaller primes. What's next?<br /><br />71 is expressible as the sum of successive composite numbers in two ways, 22 + 23 + 24 = 71 = 35 + 36 There are no smaller numbers for which this is true. It is also the sum of three consecutive primes, 19 + 23 + 29<br /><br />A 71-digit prime is formed by intertwining the even (from 2 to 40) and the odd (from 1 to 39) numbers (214365...374039) *Prime Curios By my calculation the only smaller such prime is the four digit 2143<br /><br /> 71 is the largest prime p that humans will ever discover such that 2<sup>p</sup> doesn't contain the digit 9. *Cliff Pickover (I do wonder how they go about proving such facts.)<br /><br />The sum of the prime numbers up to 71 is 639=9*71<br /><br />The smallest prime that remains prime when inserting one, two, three, or four zeros between each digit. *Prime Curios So 701, 7001, 70001, and 700001 are all prime. Students might search for numbers that can include one zero, or two, etc.<br /><br /> The Monster Group is the largest sporadic simple group and contiains appx 8 x 10^53 elements. Grant Sanderson or pointed out that its largest factor is 71. The others form a stange patter with each succesive larger factor having a lower than or equal exponent to the previous (except for one strange counterexample that makes it even more weird. \( 2^{46} x 3^{20} x 5^9 x 7^5 x 11^2 ^ 13^3 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71\) <hr /><b>The 72nd Day of the Year:</b><br />72 is a <a href="http://pballew.net/arithme2.html#pronic" target="_blank">pronic</a>, heteromecic, or oblong number (and sometimes pronic is spelled promic). They are numbers that are the product of two consecutive integers Oblong numbers have the property that if they are used in infinite nested radicals, they produce an integer, \(\sqrt(72+\sqrt(72+\sqrt(72+...))) = 9 \)<br /><br />72 is the smallest number whose fifth power is the sum of five smaller fifth powers: \(19^5 + 43^5 + 46^5 + 47^5 + 67^5 = 72^5\).<br /><br />The rule of 72 was once a commonly used approximation in banking and finance for the time it took an investment to double at r%. For a 5% investment, the approximate period would be 72/5 = 14.4 years. The rule applies to compound interest. The rule is based on an approximation of ln(2) = .693.. <br /><br />In a plane, the regular pentagon has exterior angles of 72<sup>o</sup><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-GiA-Qe-AQxE/VdzJhjY22_I/AAAAAAAAG34/a-uEQtWTxxg/s1600/rhombocuboctahedron.jpeg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-GiA-Qe-AQxE/VdzJhjY22_I/AAAAAAAAG34/a-uEQtWTxxg/s200/rhombocuboctahedron.jpeg" width="200" /></a></div>The Rhombicuboctahedron or Great Rhombicuboctahedron is an Archimedian solid that has 72 edges. It has 12 faces that are squares, 8 faces that are hexagons, and six faces that are octagons, for a total of 26 faces in all. Knowing the number of edges and faces, good students can calculate the number of vertices using <a href="http://pballew.blogspot.com/2009/04/eulers-theorem-of-planer-graphs.html" target="_blank">Euler's Gem</a>. (there is a lesser Rhombicuboctahedron or just Cubicuboctahedron which is a faceted version of the greater)<br /> <b><br /></b>72 the sum of four consecutive primes (13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19). Good plane geometry students know that the exterior angles of a regular pentagon measure 72 degrees each.<br /><br />72 is 2<sup>3</sup>+ 3<sup>2</sup>. Is is the smallest such number where the numbers are distinct primes. <br /><br /><span face="sans-serif" style="background-color: white; color: #222222;">72 is the smallest number whose fifth power is the sum of five smaller fifth powers: 19</span><sup style="background-color: white; color: #222222; font-family: sans-serif; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #222222;"> + 43</span><sup style="background-color: white; color: #222222; font-family: sans-serif; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #222222;"> + 46</span><sup style="background-color: white; color: #222222; font-family: sans-serif; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #222222;"> + 47</span><sup style="background-color: white; color: #222222; font-family: sans-serif; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #222222;"> + 67</span><sup style="background-color: white; color: #222222; font-family: sans-serif; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #222222;"> = 72</span><sup style="background-color: white; color: #222222; font-family: sans-serif; line-height: 1;">5</sup><span face="sans-serif" style="background-color: white; color: #222222;">.*Wik</span><br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><br /></span>In typography, point sizes are measured in 1/72 of an inch, 72-point characters are 1 inch tall. <br /><br />72 is the smallest number that can be expressed as the difference of the squares of consecutive primes in two distinct ways: {19<sup>2</sup> - 17<sup>2</sup>} and {11<sup>2</sup> - 7<sup>2</sup>}</div><div><br /></div><div>The number of integers less than 72 and relatively prime to it is 24. The same is true for the numbers 78, 84, and 90. This is the smallest set of four numbers in arithmetic sequence with the same value of Euler's phi function or totient function. The next string of four begins at 216. It also has an arithmetic difference of 6, and the repeated totient is (wait for it....) 72<br /><hr /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="http://mathsci2.appstate.edu/~sjg/simpsonsmath/pi.jpg" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="139" src="https://mathsci2.appstate.edu/%7Esjg/simpsonsmath/pi.jpg" width="196" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: xx-small;">Today, have your Pi(e) both ways!</span></td></tr></tbody></table><b>The 73rd Day of the Year</b><br />On non-leap years, the 73rd year day is Pi Day, March 14. 73 first occurs in the 299th digit of Pi, well behind its prime pair partner 71, which appears in the 39th position. <br /><br /> 73 is the alphanumeric value of the word NUMBER: 14 + 21 + 13 + 2 + 5 + 18 = 73 a prime number*Tanya Khovanova, <a href="http://www.numbergossip.com/" target="_blank">Number Gossip</a>;<br /><br />Expressing 73 as four 4's using the original rules of using only the four basic arithmetic functions with parentheses and concatenation has not been solved. It is one of the most difficult primes under any of the rule sets I know. (You can find a short history of the Four 4's problem <a href="https://pballew.blogspot.com/search?q=before+there+were+four+fours">here</a></div><div><br /></div><div>And if you are stumped, here is one way from Keith Raskin on LInkedIn:</div><div>"In the book Mathematical Bafflers, all integers from 0 to 100 are represented by 4 4’s, but square roots, exponentiation, factorials, decimals and infinite repetition (repetends of 4) are allowed.<br /><br /> 73 is expressed as [4!(sqrt(4) + sqrt(.4 repeating)] / sqrt(.4 repeating) = [48 + 2/3] / (2/3) = [146/3] x 3/2 = 73"</div><div><br /></div><div><br />73 is the largest prime day of the year so that you can append another digit and make another prime six times, 73, 739, 7393, 73939, 739391, 7393913, and 73939133.<br /><br />73 is the smallest prime that can be expressed as the sum of three cubes, 1^3 + 2^3 + 4^3 =73<br />The prime number 73 is the repunit 111 in octal (base 8) and the palindrome 1001001 in binary (base 2)*Prime Curios<br /><br /> Fans of the Big Bang Theory on TV know that Leonard refers to 73 as the "Chuck Norris of Numbers" After Sheldon points out that : 73 is the 21st prime, and it's mirror image 37 is the 12th prime. This enigma is the only known such combination.<br /> <iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/RyFr279K9TE" width="420"></iframe><br /><br />Sheldon failed to mention that 73 is also the 37th odd number. And it's interesting that this is the only known emirp pair where one is one less than twice the other.<br /><br /> And 73 is the smallest prime factor of a googol + 1 *Prime Curios<br /><br />73^3 =343= (3+4)^3<br /><br />If you want to represent all numbers as a sum of sixth powers, at times you will need to use at least six of them.<br /><br /> A good time to introduce you student's to a nice way to find many digits of pi, ( pick a relatively close apppx of pi (I'll use 2.5) and call it x, then x+ sin(x) is a better approximation, and repeating continues to give more and more digits of pi up to limits of calculator For 2.5 we get 3.098 -> 3.14157 -> 3.141592654 . (student's might be challenged for why (and when) this works).<br /><br />Thomas Jefferson ran against Aaron Burr for president, and they both got 73 votes.<br /><br />The sum of the first 73 odd primes, is divisible by 73<br /><br />73 =8^2 + 8^1 + 8^0<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">The smallest prime that divides a 7-digit number of the form </span><i style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif; font-size: 16px;">p</i><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">0</span><i style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif; font-size: 16px;">p</i><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">,</span><br /><br />73 is the smallest prime factor of A Googol +1<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><i>The 73rd</i> </span><b>triangular number</b><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> is equal to 73 times the reversal of 73,</span><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> </span><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">*Prime Curios </span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">73*37=2701 = the sum of the integers from 1 to 73</span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-bs-Cp32_0Rk/XsRhTbs346I/AAAAAAAAMeM/gP0jGEUMZYExHOMUqysCNGmMyinQ3vloACLcBGAsYHQ/s1600/triangle%2Bpeg%2Bsolitare.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="211" data-original-width="211" height="200" src="https://1.bp.blogspot.com/-bs-Cp32_0Rk/XsRhTbs346I/AAAAAAAAMeM/gP0jGEUMZYExHOMUqysCNGmMyinQ3vloACLcBGAsYHQ/s200/triangle%2Bpeg%2Bsolitare.jpg" width="200" /></a></div><span style="font-family: inherit;"><span style="background-color: #fcfcfc;"><span style="background-color: white; color: #222222;">A winning solution to the 15-hole triangular peg solitaire game using this method is: (4,1), (6,4), (15,6), (3,10), (13,6), (11,13), (14,12), (12,5), (10,3), (7,2), (1,4), (4,6), (6,1).</span><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">Not only does this solution leave the final peg in the original empty hole, but the sum of all the x,y hole numbers in the solution is prime (179) By the way, if you only sum the values of the landing holes, that's prime also (73).</span></span></span><br /><span style="font-family: inherit;"><span style="background-color: #fcfcfc;"><br style="background-color: white; color: #222222;" /></span> There are exactly 73 primes, beginning with the prime 1093 and ending with the prime 1613, where 1093<sup>2</sup> + 1097<sup>2</sup> + ... + 1613<sup>2</sup> = 11707<sup>2</sup>. This is the first instance of a prime number of primes comprising the left member of such an equation. </span><br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">The smallest prime that is the middle term of three consecutive numbers each expressible as a sum of two nonzero squares: 72 = 6</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">2</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> + 6</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">2</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> ; 73 = 3</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">2</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> + 8</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">2</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> ; 74 = 5</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">2</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> + 7</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">2</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">. *Prime Curios</span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span> And of course, for Pi Day, we need the world's most accurate Pi Chart <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-HZA8V_JEibs/VPkmaAsSxZI/AAAAAAAAGfU/2O-L8_76_DQ/s1600/Worlds%2Bmost%2Baccurate%2Bpi%2Bchart.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="https://1.bp.blogspot.com/-HZA8V_JEibs/VPkmaAsSxZI/AAAAAAAAGfU/2O-L8_76_DQ/s1600/Worlds%2Bmost%2Baccurate%2Bpi%2Bchart.jpg" width="320" /></a></div><hr /><b>The 74th Day of the Year</b><br />74 is related to an open question in mathematics since 74<sup>2</sup> + 1 is prime. Hardy and Littlewood conjectured that asymptotic number of elements in this sequence, primes = n<sup>2</sup> + 1, not exceeding n is approximately \(c \frac {\sqrt{n}} {log(n)}\) for some constant c. There was a $1000 prize for best solution to an open sequence during 2015 and submitting it to OEIS, details <a href="https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCIQFjAA&url=https%3A%2F%2Fwww.hs-heilbronn.de%2F7116447%2FRiordan_5.pdf&ei=EbD3VN_tMoH7sASooICICw&usg=AFQjCNEBg-NKQtogeZj4s-FevdUTK4eyDQ&sig2=PTLraq2eidlM3PO8D5gjwA&bvm=bv.87519884,d.cWc&cad=rja" target="_blank">here</a><br /><br />74 is the sum of the squares of two consecutive prime numbers. 5^2 + 7^2 Euler pointed out that any number that is twice a number that is the sum of two squares, will also be the sum of two squares.<br /><br /><br />A hungry number is number in the form 2<sup>n</sup> that eats as much pi as possible, for example 2<sup>5</sup> is the smallest power of two that contains a 3. The smallest power that contains the first three digits of pi, 314 is 2<sup>74</sup>(eating e seems much harder for powers of 2). Students might explore hungry numbers with other bases<br /><br /> 22796996699 is the 999799787th prime. Note that the sum of digits of the nth prime equals the sum of digits of n, and both sum to 74. The number 74 is the largest known digit sum with this property (as of August 2004). *Prime Curios <br /><br />There are 74 different non-Hamiltonian polyhedra with a minimum number of vertices. A Hamiltonian Polyhedra, like the Dodecahedron, is a polyhedron that has a connected circuit along the 20 vertices using distinct edges. Hamilton created a game he called the Icosian game whose object was to find the path. He used pegboard holes on the planer graph of the dodecagon. Every Platonic solid, and many others polyhedra, have a Hamiltonian cycle<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-Ep4jhT6RV2M/XkRLVssKfZI/AAAAAAAALNo/2R9QjHYUAfkuzg7lLpxY7OhmfIa4PTFpQCLcBGAsYHQ/s1600/Hamilton%2Bcircuit%2Bon%2BDodecahedron.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="206" data-original-width="202" src="https://1.bp.blogspot.com/-Ep4jhT6RV2M/XkRLVssKfZI/AAAAAAAALNo/2R9QjHYUAfkuzg7lLpxY7OhmfIa4PTFpQCLcBGAsYHQ/s1600/Hamilton%2Bcircuit%2Bon%2BDodecahedron.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Wik</td></tr></tbody></table><br /><br /><hr /><b>The 75th Day of the Year: </b><br />the aliquot divisors of 75 are 1,3,5,15, and 25. Their sum is a perfect square, 49. Their product is also a perfect square, 5625. (Can you find other numbers with this property?)<br /><br />75 is also the larger of the smallest pair of betrothed (quasi-amicable) numbers. 48 and 75 are a betrothed pair since the sum of the proper divisors of 48 is 76 and 75+1 = 76 and the sum of the proper divisors of 75 is 49, with 48+1=49. (There is only a single other pair of betrothed numbers that can be a year day)<br /><br /> 75 and 76 form the first pair of adjacent numbers in base ten which are NOT a palindrome in any base \( 2 \leq b \leq 10 \)<br /><br />2<sup>75</sup> + 75 is prime<br /><br />75 is a Keith # or repfigit (75 appears in a Fibonacci-like sequence created by its digits) 7, 5, 12, 17, 29, 46, 75 ... (75 is the sixth of seven year days which are repfigits. Can you find the others?)<br /><br />If you count all the ways 4 competitors can rank in a competition, allowing for the possibility of ties, there are 75 such possible rankings. <br />These are called Fubini Numbers, named for Italian mathematician Guido Fubini.</div><div> <br />Hare (2005) has shown that any<b> odd</b> perfect number must have <b>75</b> or more prime factors.(Nelson (2006)showed that at least nine of them must be distinct.) <hr /><b>The 76th Day of the Year:</b><br />76 is an automorphic number because the square of 76 ends in 76. (5 and 6 are automorphic because 5<sup>2</sup> ends in five and 6<sup>2</sup> ends in six). There is one other two digit automorphic number (it should be easy to find) but can you find the three digit ones? <br />In general there are two that can be found for n digit numbers, but sometimes one of them doesn't have n digits, but n-1. One of the squares (that has n digits) is the square root of 5^(2^n) where n is the number of digits, divided by 10^n. so for three digits 5^8 is the square of one of the numbers, (625) and the other is 10^3 + 1 - this first result, or 1001 - 625 = 376. Alas both solutions are too big to be year dates. <a href="https://www.johndcook.com/blog/2016/02/15/curious-numbers/">John Cook has some information with feedback f</a>rom followers that explains more. The relation between automorphic number and powers of ten is interesting. An n digit automorphic number A times (A-1) will always be a multiple of 10^n. 76 x 75 for example, is 5700. 376 x 375 = 141000. <br /><br />If you omit numbers that terminate in zero, the only multidigit numbers that have a square that ends in the original number must end in 25 or 76. \( 76^2 = 5776\), \( 376^2 = 141376\) , \(9376^2 = 87909736\) </div><div><br />76= 8 + 13 + 21 + 34 the sum of four consecutive Fibonacci numbers<br /><br />76 is the number of 6 X 6 symmetric permutation matrices.<br /><br />Seventy Six is an unincorporated community in Clinton County, Kentucky, United States. Seventy Six is 6.9 miles north of Albany( and 46 miles west of 88, ky.). Its post office has been closed.<br /><b><br /></b> <b>76 </b>can be partitioned into distinct prime integers in 76 ways. There are no other such numbers. *Prime Curios. (3 + 73 for example, is one such partit</div><div><br /></div><div>and, of course, 76 Trombones Led the Big Parade<br /><hr /><b>The 77th Day of the Year</b><br />77 is the only number less than 100 with a multiplicative persistence of 4. Can you find the next? (Multiply all the digits of a number n, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence, and the final digit obtained is called the multiplicative digital root of n.) There is not another year day that will have a multiplicative persistence greater than four. [7x7=49, 4x9=36, 3x6=18, 1x8=8]<br /><br /> 77<sup>2</sup> is the smallest square number that can be the sum of consecutive squares greater than 1, \(sum_{k=18}^{28}k^2 = 77^2\) 77^2 = 5929, the concatenation of two primes.<br /><br />The concatenation of all palindromes from one up to 77 is prime.<br /><br />77 is equal to the sum of three consecutive squares, \(4^2 + 5^2 + 6^2= 77\) and also the sum of the first 8 primes. *Prime Curios <br /><br />77 is the sum of the first eight primes, and the sum of three consecutive squares. <br /><br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">77 is the the number of digits of the 12th </span>perfect number<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">. It was discovered in 1876 by Eduoard Lucas. The largest day year that is the number of digits of a prefect number only occurs on a leap year, It is the 14th perfect number. It was discovered in 1951 by Raphael M. Robinson (1911-1995), who also discovered four others in the same year. He is perhaps more famous as the husband of Julia Robinson.</span><br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><br /></span><br /><div data-adtags-visited="true" style="background-color: white; border: 0px; color: #444444; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 14px; line-height: 1.71429; margin-bottom: 1.71429rem; padding: 0px; vertical-align: baseline;">Shortly after John D. Cook published an article on Stewart's Cube, (see Day 83) he got a response from Austin Buchanan who lowered the vertex sums to 77. </div><div data-adtags-visited="true" style="background-color: white; border: 0px; color: #444444; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 14px; line-height: 1.71429; margin-bottom: 1.71429rem; padding: 0px; vertical-align: baseline;">" I wondered if Stewart’s cube achieved its structure in the cheapest way. I considered two objectives: (1) minimize the sum of the edges incident to a vertex, and (2) minimize the weight of the worst edge. In both cases, the following cube is optimal.</div><div data-adtags-visited="true" style="background-color: white; border: 0px; color: #444444; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 14px; line-height: 1.71429; margin-bottom: 1.71429rem; padding: 0px; vertical-align: baseline;"><a href="https://farkasdilemma.files.wordpress.com/2013/08/newcube.png" style="border: 0px; color: #9f9f9f; margin: 0px; outline: none; padding: 0px; vertical-align: baseline;"><img alt="Can we call it Buchanan's cube?" class="size-medium wp-image-425 aligncenter" data-attachment-id="425" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":""}" data-image-title="newcube" data-large-file="https://farkasdilemma.files.wordpress.com/2013/08/newcube.png?w=504" data-medium-file="https://farkasdilemma.files.wordpress.com/2013/08/newcube.png?w=300" data-orig-file="https://farkasdilemma.files.wordpress.com/2013/08/newcube.png" data-orig-size="504,482" data-permalink="https://farkasdilemma.wordpress.com/2013/08/27/a-minimal-version-of-stewarts-cube/newcube/" height="286" sizes="(max-width: 300px) 100vw, 300px" src="https://farkasdilemma.files.wordpress.com/2013/08/newcube.png?w=300&h=286" srcset="https://farkasdilemma.files.wordpress.com/2013/08/newcube.png?w=300&h=286 300w, https://farkasdilemma.files.wordpress.com/2013/08/newcube.png?w=150&h=143 150w, https://farkasdilemma.files.wordpress.com/2013/08/newcube.png 504w" style="border-radius: 3px; border: 0px; box-shadow: rgba(0, 0, 0, 0.2) 0px 1px 4px; clear: both; display: block; height: auto; margin: 0.857143rem auto; max-width: 100%; padding: 0px; vertical-align: baseline;" width="300" /></a></div><div data-adtags-visited="true" style="background-color: white; border: 0px; color: #444444; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 14px; line-height: 1.71429; margin-bottom: 1.71429rem; padding: 0px; vertical-align: baseline;">It improves the first objective from 83 to 77, and the second objective from 61 to 53. To solve the problems, I modeled them as integer programs, and used Gurobi as solver. I’m currently running code for higher-dimensional cubes."</div><div data-adtags-visited="true" style="background-color: white; border: 0px; color: #444444; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 14px; line-height: 1.71429; margin-bottom: 1.71429rem; padding: 0px; vertical-align: baseline;"><br /></div><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"></span>77! + 1 is prime. 77 is the 9th year day for with this attribute, but it is the first composite number for which this is true.<br /><br />77 is the largest number that cannot be written as the sum of distinct numbers whose sum adds to one.<br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><br /><span style="color: black; font-size: medium;">Ron Graham proved that every number greater than or equal to 77 , can be partitioned into a sum of positive integers so that the sum of their reciprocals is one. It was Graham's doctoral thesis, which he called The 77 Theorem. </span></span></div><div><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><span style="color: black; font-size: medium;"><br /></span></span></div><div><span face="sans-serif" style="background-color: white; color: #222222;">It is possible for a </span>sudoku<span face="sans-serif" style="background-color: white; color: #222222;"> puzzle to have as many as 77 givens, yet lack a unique solution. </span><br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><br /></span>77 and 78 form the fourth Ruth-Aaron pair, named for the number of home runs hit by Babe Ruth, 714, and the number when Aaron broke the record, 715 (he hit more afterward). They are consecutive numbers that have the same sums of their prime factors (77 = 7*11, 78 = 2*3*13, and 7+11 = 2+3+13).</div><div><br /></div><div><br /><hr /><b>The 78th Day of the Year</b><br />78 is the smallest number that can be written as the sum of 4 distinct squares in 3 ways. *What's Special About This Number<br /><br />78 is the sum of the first twelve integers, and thus a triangular number.<br /><br />The cube of 78 is equal to the sum of three distinct cubes, 78<sup>3</sup> = 39<sup>3</sup> + 52<sup>3</sup> + 65<sup>3</sup><br />(Historically, it seems Ramanujan was inspired by a much smaller such triplet 6<sup>3</sup> = 3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup><br /><br />77 and 78 form the fourth Ruth-Aaron pair, named for the number of home runs hit by Babe Ruth, 714, and the number when Aaron broke the record, 715 (he hit more afterward). They are consecutive numbers that have the same sums of their prime factors (77 = 7*11, 78 = 2*3*13, and 7+11 = 2+3+13).<br /><br />78, is a sphenic number, having 3 distinct prime factors. (A good word for students to learn) 78=2*3*13<br /><br />78 is the 12th Triangular number, the sum of the digits from 1 to 12.<br /><br />78 is a palindromic number in bases 5 (303<sub>5</sub>, and base 7 (141<sub>>7</sub> (77 which is a palindrome itself, is not a palindrome in any smaller base.<br /><br />78 is the number of cards in a tarot deck containing the 21 trump cardsand the 56 suit cards. *Wik<br /><br /><span style="background-color: white;">Ron Graham proved that every number greater than or equal to 77 , can be partitioned into a sum of positive integers so that the sum of their reciprocals is one. 78 = 2 + 6 + 8 + 10 + 12 + 40, and 1/2 + 1/6 + 1/8 + 1/10 + 1/12 + 1/40 = 1</span><br /><hr /><b>The 79th Day of the Year:</b><br /><b><br /></b><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">79 is the smallest </span>prime<span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> whose sum of digits is a fourth power. *Prime Curios (</span>the second smallest is its reversal, 97)<b><br /></b><b><br /></b><br /> 78*79 = 6162 (note that the product of consecutive numbers produces a number that is the concatenation of two successive numbers 61 and 62 in ascending order (and 61 is prime). (<i>Can you find another number, not necessarily prime, so that n(n-1)= a concatenation of consecutive numbers?</i>)<br /><br />79 = 2<sup>7</sup> - 7<sup>2</sup><br /><br />79 is the smallest number that can not be represented with less than 19 fourth-powers. (Before you read blythly on, there are three more year days that also require the sum of 19 fourth-powers... find one.)<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">Each of the numbers 1 to 79 gives a larger number when you write out its English name and add the letters using a=1, b=2, c=3, ... (but 80 gives 74)*Prime Curios</span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span>79 = 11 + 31 + 37. Curiously, the sum holds for the reversals: 97 = 11 + 13 + 73, and all are primes.<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">On page 79 of the novel </span><i style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif; font-size: 16px;">Contact</i><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> by Carl Sagan, it says that no astrophysical process is likely to generate prime numbers.*Prime Curios</span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span>2<sup>79</sup> is the smallest power of 2 which is greater than Avogadro's number<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">79 is the largest number as the sum of the product of two successive primes. </span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span>10<sup>79 </sup>has been called the "Universe number" because it is considered a reasonable lower limit estimate for the number of atoms in the observable universe. *<a href="http://amzn.to/1XiwxgB" target="_blank">Prime Curios</a><br /><b><br /></b> 79 is an Emirp, a prime whose digit reversal is another prime. <span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">and which is the sum of three other Emirps, and also true for the sum of their reversals 79 = 11 + 31 + 37and </span><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> 97 = 11 + 13 + 73.</span><br /><span face=""comic sans ms" , "georgia" , sans-serif"><br /></span><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">2</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">79</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> = 604462909807314587353088 is the smallest pandigital number of the form 2 to the power of prime.*Prime Curios</span><br /><span face=""comic sans ms" , "georgia" , sans-serif"><br /></span><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">There are 79 ways to place four non-attacking chess kings on a 4 × 4 board. </span><br /><br />79 square cm is the typical surface area of an average brick. *Jennifer Li</div><div><br /><hr /><b>The 80th Day of the Year: </b><br />There are 80 four-digit primes which are concatenations of two-digit primes. (<i>3137 is one example, can you find the rest?</i>) *<a href="http://primes.utm.edu/" target="_blank">Prime Curios!</a><br /><br />80 in Roman Numerals is not suitable for minors, LXXX,<br /><br />The Pareto principle (sometimes called the 80-20 rule) says that, for many events, roughly 80% of the effects come from 20% of the causes, ie, \( \approx 80\% \) of the accidents are caused by 20% of the drivers.<br /><br />\(n*2^{n-1} \) gives the number of edges (segments) in a n-dimensional cube, and in the 5th dimension, (went there once in a dream) there are 80 edges, 5*2<sup>4</sup><br />(It also has eighty two-dimensional square faces.)<br /><br />And 80 is the <b>smallest</b> number diminished by taking its sum of letters (writing out its English name and adding the letters using a=1, b=2, c=3, ...) *Tanya Khovanova<br /><br />In 1719 Paul Halcke showed that the product of the aliquot divisors of 80 equals the fourth power of 80. The only year numbers for which this is true is 48 and 80.<br /><br />80 is the smallest integer n such that both n and n+1 are products of at least 4 primes.*Prime Curios <br /><br />80 is a semiperfect number, since adding up some subsets of its divisors 1 + 4 + 5 + 10 + 20 + 40 gives 80.<br /><br />80 is the largest natural number n such that all prime factors of n and n+1 are smaller or equal to the prime digit 5.<br /><br />80 is palindromic in bases 3, 6, and 9 , and a Repdigit in base 3 and base 9.<br /><hr /><b>The 81st Day of the Year:</b><br />There are many numbers for which if you raise the sum of the digits to some power you get the original number. 512 for instance, (5+1 + 2) ^ 3 = 8^3 = 512. For 2401 we can have (2+4+0+1)^4 = 9^4 = 2401. For year dates, the only such numbers that work are 1, and 81. 81 is the largest known number such that the square of the sum of its digits is equal to the number. ( 8 + 1) ^2 = 9^2 = 81. <br /><br /><br />Student's are reminded that 8 x 10 + 1 is 81 = 9^2, an example of a beautiful mathematical truth that 8 x T + 1 is a square for any triangular number. <br />The smallest 10 digit pandigital number is 1023456789, 81 or 3<sup>4</sup> is a factor. The other two factors are both four digit numbers. Can you find the smallest, which is prime? ;-} <br /><br />81 is one of only three non-trivial numbers for which the sum of the digits * the reversal of the sum yields the original number (8+1 = 9; 9*9 = 81). The other two are the famous Hardy-Ramanujan taxicab # 1729, which is the smallest number which is the sum of two positive cubes in two ways.(1+7+2+9 = 19; 19*91 = 1729), and 1458 (1+4+5+8 = 18; 18*81=1458) which is also unique for being the maximum determinant possible for a 11x11 matrix with only ones and zeros. The Hardy-Ramanujan number's properties were first noted by Frénicle de Bessy in 1657(without mention of a taxicab).That is, he knew it was the sum of two cubes in two ways, not sure if he knew the sum of the digits property above... anyone?<br /><br />"A dozen, a gross and a score,<br />Plus three times the square root of four,<br />Divided by seven,<br />Plus five times eleven,<br />Is nine squared, and not a bit more. "<br />John Saxon.<br />(12+ 144 + 20 + 3*sqrt(4)) /7 + 5 *11 = 9^2 = 81.<br /><br />80 and 81 are the smallest integer pair of consecutive numbers such that both are products of at least 4 primes.*Prime Curios<br /><br />and 80 and 81 are the largest integer pair such that all prime factors of the two are smaller than or equal to the prime digit 5.<br /><br />81 is the smallest square (and only known) such that n*2<sup>n</sup>-1 is prime *Prime Curios<br /><iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=httppbalnet-20&marketplace=amazon&region=US&placement=1448651700&asins=1448651700&linkId=9261c7c6a47d13d1589e251a650dc426&show_border=true&link_opens_in_new_window=true" style="height: 240px; width: 120px;"></iframe><br /><br />There are 81 Full House primes, with the most advantageous being 18181 and 81181 (where 1 is regarded as the Ace.) Supposedly the famous frontier hero, Wild Bill Hickock, died by being shot from behind in a poker game. His hand was a supposedly two pairs, with Aces and Eights and is often called the "dead man's hand". A full house is three of one card and a pair of another. <br /><br />The decimal expansion of the reciprocal of 81 is \( \overline{ .012345679}.... \) omitting only the 8. This is true of the reciprocal of (b-1)<sup>2</sup> in any base b leaving out only the value b-2 in the repeating period. For example, in base 5, the reciprocal of 16 (31<sub>5</sub> ) is \( \overline{.0124...} \)<br /><br />If you look into the palm of your left hand with your fingers extended parallel to the ground, you will see the Arabic numeral for eight near the center of your palm, and the numeral for one just above it.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-WWKpo5FHkl8/Xkhx5RrR7rI/AAAAAAAALOM/lWa_112WonAS61zYWAEKLMHwIMEQJOYGQCLcBGAsYHQ/s1600/arabic%2Bnumberals.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="119" data-original-width="400" height="94" src="https://1.bp.blogspot.com/-WWKpo5FHkl8/Xkhx5RrR7rI/AAAAAAAALOM/lWa_112WonAS61zYWAEKLMHwIMEQJOYGQCLcBGAsYHQ/s320/arabic%2Bnumberals.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Facebook</td></tr></tbody></table><br />81 is the third smallest number where the sum of its divisors is a perfect square. \(1 + 3 + 9 + 27 + 81 = 11^2 = 121\) There are only three year dates (to my knowledge) for which the sum of the divisors is a square. They all three occur in this MathFacts page, 66, 70, and 81. 66 and 70 even have the same total, 144. The largest of the three, 81, has the smallest sum. </div><div><br /></div><div><br /></div><div><hr /><b><br /></b><b>The 82nd Day of the Year:</b><br />82 is the sum of the 10th(8+2) prime and the 16th(8x2) prime. It is the smallest number with this property. Can you find the next?<br /><br />82 is a happy number. Take the sum of the square of the digits, repeat on the result, and you eventually arrive at 1.<br /><br /> 82 is the number of different ways you can arrange 6 regular hexagons by joining their adjacent sides: <br /><div class="separator" style="clear: both; text-align: center;"></div><a href="https://1.bp.blogspot.com/-fNMdXGUhpps/VugmfPEdbVI/AAAAAAAAIEs/7p87GfXD-xk3NacuMLf_Rgo6JoCbfhM4g/s1600/82-hexacomb.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="https://1.bp.blogspot.com/-fNMdXGUhpps/VugmfPEdbVI/AAAAAAAAIEs/7p87GfXD-xk3NacuMLf_Rgo6JoCbfhM4g/s400/82-hexacomb.gif" width="400" /></a><br /><br /> 82 can be written as : The sum of Fibonacci numbers, 82 = 1 + 5 + 21 + 55 The sum of consecutive integers, 82= 19 + 20 + 21 + 22 and as the sum of squares 82= 1<sup>2</sup> + 9<sup>2</sup> *What's Special About This Number<br /><br />82 is a palindrome in base 3(1001) and in base 9 (101) (compare to 80 which is a palindrome in both 3 and 9, as well as 6).<br /><br />82 is a companion Pell Number."<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">In mathematics</span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, the </span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">Pell numbers</span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> are an infinite </span>sequence<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> of </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Integers" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Integers">integers</a><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, known since ancient times, that comprise the </span>denominators<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> of the </span>closest rational approximations<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> to the </span>square root of 2<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">. This sequence of approximations begins </span><br /><span class="sfrac nowrap tion" face="sans-serif" role="math" style="background-color: white; color: #222222; display: inline-block; font-size: 11.9px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">1</span><span class="slash visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">1</span></span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, </span><span class="sfrac nowrap tion" face="sans-serif" role="math" style="background-color: white; color: #222222; display: inline-block; font-size: 11.9px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">3</span><span class="slash visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, </span><span class="sfrac nowrap tion" face="sans-serif" role="math" style="background-color: white; color: #222222; display: inline-block; font-size: 11.9px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">7</span><span class="slash visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">5</span></span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, </span><span class="sfrac nowrap tion" face="sans-serif" role="math" style="background-color: white; color: #222222; display: inline-block; font-size: 11.9px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">17</span><span class="slash visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">12</span></span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, and </span><span class="sfrac nowrap tion" face="sans-serif" role="math" style="background-color: white; color: #222222; display: inline-block; font-size: 11.9px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">41</span><span class="slash visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">29</span></span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">companion Pell numbers</b><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> or </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">Pell–Lucas numbers</b><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82."*Wik</span><br />Both integer sequences are formed by multiplying twice the last member and adding the one before that, so the companion number 82 = 2*34+14. The associated denominator, or Pell number, 29=2*17 + 7.</div><div><br /></div><div>Physicist Otto Stern was nominated for the Nobel Prize on a total of 82 occasions, ultimately receiving the 1943 Physics Prize. Among his nominators were Albert Einstein, Wolfgang Pauli, Max Planck and Werner Heisenberg.<br /><hr /><b>The 83rd Day of the Year</b><br />83 is the smallest prime whose square, 6889, is a strobogrammatic number. <br /><br />83 is the sum of the squares of the first three consecutive odd primes (3^2 + 5^2 + 7^2).<br /><br />One of my Mathematical idols, Paul Erdos, died at age 83. I heard about it when one of my advanced math students announced, 'Mr Ballew, you know that Math guy you always talk about; I heard he died last night."<br /><br />83 is the smallest prime number which is the sum of a prime number of consecutive prime numbers in a prime number of different ways, i.e., 23 + 29 + 31 = 11 + 13 + 17 + 19 + 23. *Prime Curios (<i>Whew! say that three times in a hurry</i>)<br /><br />The smallest prime with a digit sum of 83 is 3999998999.<br /><br />83 is The number of permutations of the 10 distinct digits taken 9 at a time that are perfect squares. These range from 10124<sup>2</sup> = 102495376 to 30384<sup>2</sup> = 923187456.*Prime Curios<br /><br /> Stewart's cube, shown below, is a graph with 8 vertices and 12 edges. Each edge is assigned a "weight" of unique prime numbers, and the total weight of the three edges meeting at each vertex is 83. <br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-d4i2_lCra-s/WbrdJuS2WwI/AAAAAAAAI0I/ATaTMHyXUuwKEhb0Hw_tDosBQMCdaXgKgCLcBGAs/s1600/stewarts_cube.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="482" data-original-width="504" height="306" src="https://3.bp.blogspot.com/-d4i2_lCra-s/WbrdJuS2WwI/AAAAAAAAI0I/ATaTMHyXUuwKEhb0Hw_tDosBQMCdaXgKgCLcBGAs/s320/stewarts_cube.png" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*John D Cook</td></tr></tbody></table>As far as I know this is the first graph found with these properties (unique prime edges and common total). I would like information on which "Stewart" this is named for and when. There is a more recent discovery by <a href="https://farkasdilemma.wordpress.com/2013/08/27/a-minimal-version-of-stewarts-cube/" target="_blank">Austin Buchanan</a> that has all prime edges and a smaller weight of 77 at each vertex. <br /><br />83 is a prime number of the form 4n-1 so it can not be expressed as the sum of two squares.<br /><br />And I just found out that in the Jewish faith, when you reach the age of 83, you can celebrate a 2nd Bar Mitzvah.<br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><br /></span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">"The </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">TI-83 series</b><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> is a series of </span><a href="https://en.wikipedia.org/wiki/Graphing_calculator" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Graphing calculator">graphing calculators</a><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> manufactured by </span><a href="https://en.wikipedia.org/wiki/Texas_Instruments" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Texas Instruments">Texas Instruments</a><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">. The original TI-83 is itself an upgraded version of the </span><a href="https://en.wikipedia.org/wiki/TI-82" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="TI-82">TI-82</a><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">.</span><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"> Released in 1996, it was one of the most popular graphing calculators for students. In addition to the functions present on normal </span><a href="https://en.wikipedia.org/wiki/Scientific_calculator" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Scientific calculator">scientific calculators</a><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">, the TI-83 includes many features, including function graphing, polar/parametric/sequence graphing modes, statistics, trigonometric, and algebraic functions, along with many useful </span><a href="https://en.wikipedia.org/wiki/Application_software" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Application software">applications</a><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">. Although it does not include as many calculus functions, applications and programs can be downloaded from certain websites, or written on the calculator." Wik. </span><br /><hr /><b>The 84th Day of the Year:</b><br />with nine points equally placed around a circle there are 84 different triangles using three of these points as vertices. <i>Are any of them right triangles? </i><br /><br />And Jim Wilder hit the trifecta today with :<br />\(84 = 2^5 + 3^3 + 5^2 \)<br />\(84 = 2^2 + 4^2 + 8^2 \) and<br />\( 84 = 1^3 + 1^3 +1^3 + 3^3 + 3^3 + 3^3 =3^3+3\)<br /><br />And 84 is the sum of three consecutive powers of four \(4^1 + 4^2 + 4^3 = 84\)<br />84 is the only number that is spelled with ten letters that are all different.<br /><br />84 is the sum of the first seven triangular numbers, making it the 7th Tetrahedral number.<br /><br />84 is the smallest number that can be expressed as the sum of three distinct primes raised to prime exponents, 2^5 + 3^3 + 5^2 = 84<br /><br />84 is also the smallest number that can be expressed as the sum of two primes in 8 different ways. 5+79 is one, the rest are on you.<br /><br />84 can also be written in four different ways as the sum of four primes, with any three of them summing to a prime. 5 + 13 + 23 + 43 is one of them, *Prime Curios<br /><br />Little is known of the life of Diophantus, but this problem, supposedly on his tomb, will reveal his age. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-rXdKnNA2wkI/VuxrBEWIkqI/AAAAAAAAIG4/W5z8VuezkHQs9pyl2g7T1BLFOOrBSGf3Q/s1600/Diaphontus%2Blife%2Bpuzzle.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="164" src="https://4.bp.blogspot.com/-rXdKnNA2wkI/VuxrBEWIkqI/AAAAAAAAIG4/W5z8VuezkHQs9pyl2g7T1BLFOOrBSGf3Q/s640/Diaphontus%2Blife%2Bpuzzle.jpg" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">84 is a Hoax number, the sum of it's digits is the same as the sum the digits of it's unique prime factors. It is the third year day which is a hoax number. \( 84 = 2^2 x 3 x 7 \) . So unique prime factors are 2, 3, and 7 with a sum of 12. 8+4 = 12 also. <span style="text-align: center;">Tomorrow will be the fourth. </span> </div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">And 84 is still, I believe, the largest number of times anyone has been nominated for the Noble Prize and never won even once. During the 1901-1950 period, Arnold Sommerfeld was nominated for the Nobel Prize 84 times, more than any other physicist (including Otto Stern,who got nominated 81 times), but he never received the award. His PhD students also earned more Nobel prizes in physics than any other supervisor’s,ever. *Wik </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"> Eighty Four is a census-designated place in Somerset, North Strabane, North Bethlehem and South Strabane townships in Washington County, Pennsylvania, United States. It lies approximately 25 miles (40 km) southwest of Pittsburgh and is in the Pittsburgh metropolitan area. The population was 657 at the 2010 census.</div><hr /><b>The 85th Day of the Year:</b><br />85 is the largest number for which the sum of 1<sup>2</sup> + 2<sup>2 </sup>+ 3<sup>2 </sup>+ 4<sup>2 </sup>+...+n<sup>2</sup>= 1+2+3+4+.... +M for some n,M, can you find that M? 85 is the largest such n, with a total sum of squares of 208,335; but can you find some solutions n,M that are smaller? (<i>Reminder for students, the sums of first n consecutive squares are called pyramidal numbers, the sums of the first n integers are called triangular numbers.)</i><br /><br />And a bonus I found at the <a href="http://primes.utm.edu/curios/page.php?short=85" target="_blank">Prime Curios </a>web site, (85<sup>11</sup> - 85)/11 ± 1 are twin primes.<br /><br />85, with 86 and 87 are the second smallest three consecutive numbers such all three are products of two primes. *Don S McDonald <br /><br />85 is the fourth Hoax number of the year. Yesterday was the third. A Hoax number is a number with the sum of it's digits equal to the sum the digits of it's unique prime factors.<br /><br />There are 85 five-digit primes that begin with 85.<br /><br />85 is the sum of consecutive integers, and the difference of their squares \(42+43= 43^2 - 42^2 = 85\),<br /><br />85 is the ONLY known Smith Number(sum of its digits is same as the sum of the digits of its prime factorization), whose aliquot sum(sum of all prime numbers less than n) is equal to \( \pi(n)\) (The number of primes less than, or equal to n)<br />85 can be expressed as the sum of two squares in two different ways, 9<sup>2 </sup>+ 2 <sup>2</sup> = 7<sup>2</sup> + 6<sup>2</sup> =85. It is the smallest such number with all squares greater than one.<br /><br />85 is the length of the hypotenuse in four Pythagorean Triangles. @igor_goldshtein gave the legs, 13 & 84, 36 & 77, 40 & 75, and 51 & 68. <br /><br />The only composite number resulting from the sum of three double-digit consecutive emirps, 17, 31, 37. Note that the concatenation in order of these emirps, i.e., 173137 is prime *Prime Curios<br /><br />85 is a palindrome in base 2 (1010101) and a palindrome and a repdigit in base 4 (1111) (85=1+4+16+64)<br /><br /><iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=httppbalnet-20&marketplace=amazon&region=US&placement=0140261494&asins=0140261494&linkId=f15dad2ac0d549ca8f45a6ff305cb4da&show_border=true&link_opens_in_new_window=true" style="height: 240px; width: 120px;"></iframe><br /><br /><hr /><b>The 86th Day of the Year</b><br />86 is conjectured to be the largest number n such that 2<sup>n</sup> (in decimal) doesn't contain a 0. *Tanya Khovanova, Number Gossip<br /><br />The 86th prime is 443, and 443<sup>3</sup> = 86,938,307. There is no other two digit n, such that the nth prime starts with n.<br /><br />86 is the sum of four consecutive integers, 86= 20 + 21 + 22 + 23 and of four consecutive squares,<br /><br />85, with 86 and 87 are the second smallest three consecutive numbers such all three are products of two primes. *Don S McDonald<br /><br />86= 3<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> This is a Truncated Pyramidal number, calculating the sum of the squares from n, to 2n. There are only five such year days, 5, 29, 86, 190 and 355.<br /><br />The multiplicative persistence of a number is the number of times the iteration of finding the product of the digits takes to reach a one digit number. For 86, with persistence of three, we produce 8*6= 48, 4*8 = 32, and 3*2 = 6.... and 48+32+6 = 86. (how frequently does that occur?)<br /><br />There are 86 abundant numbers(the sum of the proper divisors is greater than the number) in a non-leap year, but 86 is not one of them. All the abundant year days are even numbers. The smallest odd abundant number is 945.<br /><br />86 in base 6 is a repdigit, and thus a palindrome (222) that makes it equal to twice the digits of the beast number raised to consecutive power, 86 = 2* (6^0 + 6^1 + 6^2)<br /><br />Kentucky Highway 86 is only a fraction of a mile greater than 86/2. It runs from Union Star to US 62 near Cecillia. <br /><hr /><b>The 87th Day of the Year:</b><br />87 is the sum of the squares of the first four primes is 87. \(87 = 2^2 + 3^2 + 5^2 + 7^2 \)<br /><br />87 = 3 * 29, \(87^2 + 3^2 + 29^2 and 87^2 - 3^2 - 29^2 \)are both primes<br /><br />Among Australian cricket players, it seems, 87 is an unlucky score and is referred to as "the devil's number", supposedly because it is 13 runs short of 100.<br /><br />87 is the third consecutive day that is semiprime (the product of two primes), 85, with 86 and 87 are the second smallest three consecutive numbers such all three are products of two primes. *Don S McDonald<br /><br />And 87 is, of course, the number of years between the signing of the U.S. Declaration of Independence and the Battle of Gettysburg, immortalized in Abraham Lincoln's Gettysburg Address with the phrase "fourscore and seven years ago..."<br /><br />87 is the largest number that yields a prime when any of the one-digit primes 2, 5 or 7 is inserted between any two digits. The only other such number is 27 (and trivially, the 1 digit numbers). *Prime Curios <br /><br />5! - 4! - 3! - 2! - 1! = 87. Remember the old puzzle of making numbers with four 4's. What numbers could you make with the first five factorials using only the four basic arithmetic functions between them<br /><br /><hr /><b>The 88th Day of the year:</b><br />88<sup>2</sup> = 7744, it is one of only 5 numbers known whose square has no isolated digits. (<i>Can you find the others?</i>) [Thanks to Danny Whittaker @nemoyatpeace for a correction on this.] I find it beautiful that a </div><div>two digit repeating number, has a square that is a concatenation of two such numbers. <br /><br />There are only 88 narcissistic numbers in base ten, (an n-digit number that is the sum of the nth power of its digits, 153=1<sup>3</sup> + 5<sup>3</sup> + 3<sup>3</sup><br /><br />88 is also a chance to introduce a new word (It was new to me). 88 is <b>strobogrammatic</b>, a number that is the same when it is rotated 180<sup>o</sup> about its center... 69 is another example. (Not everyone holds to these rules, even me) If they make a different number when rotated, they are called invertible (89 becomes 68 for example). *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a><br /><br /> <iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=httppbalnet-20&marketplace=amazon&region=US&placement=1448651700&asins=1448651700&linkId=a39b21fd7e0b74cd67b7bcdaa27cf5ee&show_border=true&link_opens_in_new_window=true" style="height: 240px; width: 120px;"></iframe><br /><br /> And with millions (billions?) of stars in the sky, did you ever wonder how many constellations there are? Well, according to the Internationals Astronomical Union, there are 88.<br /><div class="_uX kno-fb-ctx" data-hveid="28" data-ved="0ahUKEwiR3dH_wNDLAhUJzGMKHcIRALUQtwcIHCgAMAA" role="heading"><div class="_eF" data-tts-text="88" data-tts="answers"></div></div><div class="_oDd" data-hveid="29"><span class="_Tgc _y9e">Currently, 14 men and women, 9 birds, two insects, 19 land animals, 10 water creatures, two centaurs, one head of hair, a serpent, a dragon, a flying horse, a river and 29 inanimate objects are represented in the night sky (the total comes to more than <b>88</b> because some constellations include more than one creature.)</span><br /><span class="_Tgc _y9e"><br /></span></div><div class="_oDd" data-hveid="29"></div>And if you chat with Chinese friends, the cool way to say bye-bye is with 88, from Mandarin for 88, "bā ba". <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-0UNunMFnMSg/Vs4_VkTx8EI/AAAAAAAAH4U/zWqJmaAnTxc/s1600/eightyeightcoverfront.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="377" src="https://2.bp.blogspot.com/-0UNunMFnMSg/Vs4_VkTx8EI/AAAAAAAAH4U/zWqJmaAnTxc/s640/eightyeightcoverfront.jpg" width="640" /></a></div>Not too far from my home near Possum Trot, Ky, there is a little place called Eighty-eight, Kentucky. One strory of the naming (there could be as many as 88 of them) is that the town was named in 1860 by Dabnie Nunnally, the community's first postmaster. He had little faith in the legibility of his handwriting, and thought that using numbers would solve the problem. He then reached into his pocket and came up with 88 cents. I<span class="irc_su" dir="ltr" style="text-align: left;">n the 1948 presidential election, the community reported 88 votes for Truman and 88 votes for Dewey, which earned it a spot in Ripley's Believe It or Not. </span> <a href="https://en.wikipedia.org/wiki/Eighty_Eight,_Kentucky#cite_note-2"></a>And expanding the "88 is strobogrammatic" theme, INDER JEET TANEJA came up with this beautiful magic square with a constant of 88 that was used in a stamp series in Macao in 2014 and 2015. This image shows the reflections both horizontally and vertically, as well as the 180 degree rotation, each is a magic square. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-JPhQPNEW1jQ/VqY283_j7hI/AAAAAAAAHpY/PDZn2R6ZPOU/s1600/88%2Bmagic%2Bsquare.jpe" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-JPhQPNEW1jQ/VqY283_j7hI/AAAAAAAAHpY/PDZn2R6ZPOU/s400/88%2Bmagic%2Bsquare.jpe" width="400" /></a></div>The stamps had denominations of 1 through 9 pataca and when two sheets were printed you could do your own Luo Shu magic square with the denominations. The Luo Shu itself was featured on the 12 pataca stamp. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-NDvquVTYMlI/VqY5NoasyUI/AAAAAAAAHpk/Kl9y9B72_S4/s1600/lo%2Bshu%2Bstamp%2Bmacao.jpeg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://4.bp.blogspot.com/-NDvquVTYMlI/VqY5NoasyUI/AAAAAAAAHpk/Kl9y9B72_S4/s400/lo%2Bshu%2Bstamp%2Bmacao.jpeg" width="400" /></a></div><br />88 is called a Refactorable Number, because it is divisible by the number of divisors it has. It's the fifteenth so far this year.<br /><br />The sum of the first 88 emirps, (primes that are still prime when digit order is reversed, 13 and 31 for example) is 88,000, The 88th emirp is 1831*Prime Curios.<br /><br />88 is a palindrome in base ten, and also in base 5 (323).<br /><br />And a good reason to remind students Why 88 can not be expressed as the sum of two squares. Any number that has a factor of the form 4n-1 (for 88 that's the 11) that has an odd exponent (like 1) is not the sum of two squares.<br /><br />88 and 945 are the smallest abundant numbers that share no common divisors. 88 is the 19th abundant number and like all the numbers before it, is even. (45 is the smallest odd abundant number, following 231 even numbers. 945 has prime divisors of 3, 5, and 7 but 88 is the smallest abundant number that has none of these.</div><div><br /></div><div><span face="-apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif" style="background-color: rgba(0, 0, 0, 0.008); color: #0f1419; font-size: 15px; white-space: pre-wrap;">In the standard English counting system, every single number after EIGHTY-EIGHT has an N in its name. *HT to @HaggardHawks</span></div><div><span face="-apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif" style="background-color: rgba(0, 0, 0, 0.008); color: #0f1419; font-size: 15px; white-space: pre-wrap;"><br /></span></div><div>An Interesting pattern for 88,</div><div>9*9+7=88</div><div>98*9+6=888</div><div>987*9+5=8888</div><div>9876*9+4=88888</div><div>And it continues .. even into the negative numbers...</div><div>987654321*9-1 = 8888888888</div><div><br /><hr /><b>The 89th Day of the year:</b><br />89 is the fifth Fibonacci prime and the reciprocal of 89 starts out 0.011235... (generating the first five Fibonacci numbers) *<a href="http://primes.utm.edu/curios/home.php" target="_blank">Prime Curios</a> It actually generates many more, but the remainder are hidden by the carrying of digits from the two digit Fibonacci numbers. (The next digit, for instance is a 9 instead of an eight because it includes the tens digit of the next Fibonacci number, 13.)<br /><br />89 is the 11th Fibonacci number, and it is not divisible by any smaller Fibonacci numbers. \( F_m divides F_n \) only if m divides n. Since 89 is the 11th Fibonacci number so it is not divisible by any smaller Fibonacci number. This can help if finding possible factors for numbers are knowing if they are prime. 377 is the 14th Fibonacci number, and 14 is divisible by 7, so 377 is not prime, and is divisible by by F<sub>_7</sub> which is 13. 377 = 13 x 29.<br /><br />One more note about Fibonacci numbers that ties them to a sequence of hypotenii of right triangles, if you look at Fibonacci numbers beginning with 5; <b>5</b>, 8, <b>13</b>, 21, <b>34</b>, 55, <b>89... </b>Every other one is the hypotenuse of a Pythagorean triangle. For this case, 39, 80, 89 are the three sides. But if you look at the previous one, 16, 30, 34 you might notice that the sum of these sides is 89, and such is true for each of them. </div><div><br />89 is a factor of 2^11 - 1, the smallest factorable Mersenne Number with a prime exponent.<br /><br />89 is the 24th prime number, </div><div><br />The base of the Statue of Liberty is 89 feet tall.<br /><br />89 is an invertible number (rotation by 180 degrees produces a different number, 68) Others refer to such rotations that produce another number as Strobogrammatic. Even another term for them is numeric ambigrams. Which ever term you use, it is the sum of 1+8+11+69, four strobogrammatic numbers.<br /><br />and 89 can be expressed by the first 5 integers raised to the first 5 Fibonacci numbers: 1<sup>1</sup> + 2<sup>5</sup> + 3<sup>3</sup> + 4<sup>1</sup>+ 5<sup>2</sup><br /><br />If you write any integer and sum the square of the digits, and repeat, eventually you get either 1, or 89 (ex: 16; \( 1^2 + 6^2 = 37; 3^2 + 7^2 = 58; 5^2 + 8^2 = 89 \)<br /><br />91^2 - 89 ^2 = 360. *Gary Croft pointed out in a post that if you take the 24 numbers up to 89 that are not divisible by any of 2, 3, or 5 (he uses this set for a very efficient prime sieve) they pair up so that the difference of their squares are multiples of 360 (89^2 - 1^2 = 360 x 22; 83^2 - 7^2 = 360 x 19; ....)</div><div><a href="https://www.primesdemystified.com/twinprimes.html" target="_blank">His work is here.</a></div><div><br />89 = 8<sup>1</sup> + 9<sup>2 </sup> There are two three digit year dates that share this property, abc = a^1 + b^2 + c^3</div><div> <br />An Armstrong (or <i>Pluperfect digital invariant</i>) number is a number that is the sum of its own digits each raised to the power of the number of digits. For example, 371 is an Armstrong number since \(3^3+7^3+1^3 = 371\). There are exactly 89 such numbers, including two with 39 digits. (115,132,219,018,763,992,565,095,597,973,522,401 is the largest) (Armstrong numbers are named for Michael F. Armstrong who named them for himself as part of an assignment to his class in Fortran Programming at the University of Rochester \)<br /><br />89 is the smallest prime (indeed the smallest positive integer) whose square (7921) and cube (704969) are likewise prime upon reversal. *Prime Curios<br /><br /> And from our strange measures category, A <b>Wiffle</b>, also referred to as a WAM for Wiffle (ball) Assisted Measurement, is equal to a sphere 89 millimeters (3.5 inches) in diameter – the size of a Wiffle ball, a perforated, light-weight plastic ball frequently used by marine biologists as a size reference in photos to measure corals and other objects. The spherical shape makes it omnidirectional and perfect for taking a speedy measurement, and the open design also allows it to avoid being crushed by water pressure. Wiffle balls are a much cheaper alternative to using two reference lasers, which often pass straight through gaps in thin corals. A scientist on the research vessel EV Nautilus is credited with pioneering the technique *Wik <br /><br />The sum of all the prime numbers up to and including 89 is 963. If you eliminate the single even prime, 2, the total is 961 = 31^2. This is the smallest such sum of primes for which this is true.<br /><br />The concatenation of all odd primes starting from 89 and counting in reverse is prime.*Prime Curios (so what if it starts with 83? <br />89 is a Pythagorean Prime, one that is the sum of two squares. 8^2 + 6^2<br /><br />2<sup>2</sup> + 3<sup>3</sup> + 5<sup>5</sup> + 7<sup>7</sup> + 11<sup>11</sup> + ... + 89<sup>89</sup> is prime.<br /><br />Hellin's law states that twins occur once in 89 births, triplets once in 892 births, and quadruplets once in 893 births, and so forth. This approximation came before the advent of fertility methods. *Prime Curios <br /><br />8989<sup>2</sup> = 80802121. 89 is the smallest prime, P so that PP<sup>2</sup> is of the form XXYY, *Prime Curios<br /><br />89^2 - 2 = 7919, the 1000th prime, so there are exactly 1000 primes between 1 and 89<sup>2</sup><br /><br />89 is a palindrome in base 8 (131)<br /><br />How about some sums of sequential primes raised to their own power....that are prime!!!<br />From the clever Tweets of Srinivasa Raghava K \( 2^2 + 3^3 = 13 \) and<br />\(2^2 + 3^3 + 5^5 + 7^7 = 826699\) also prime, and<br />\(2^2 + 3^3 + \dots + 89^{89} \) is BIG... and Prime<br /><br />Oklahoma City, Ok. was founded in the Land Run of 89, when property in the Indian Nation was opened to white settlers. <br /><br />Double 89 and add 1, you get a prime. Keep doing it and you'll get a sequence of six such primes. Primes p for which 2p + 1 are also prime are called Sophie Germain primes, after a great female mathematician.<br /><br />US 89 is called the National Parks Hwy, and runs, almost literally, through Yellowstone Natl Park, and Links it to six more.<br /><hr /><b>The 90th day of the year;</b><br />90 is the <i>only</i> number that is the sum of its digits plus the sum of the squares of its digits. (<i>Is there any interesting distinction to the rest of the numbers for which this sum is more (or less) than the original number?</i>)<br /><br /> \( \frac{90^3 - 1}{90 - 1} \) is a Mersenne prime.<br /><br />90 = 3^2 + 9^2 = 3^2 + 3^4<br /><br />90 is a Harshad (Joy Giver) number since 90 is divisible by the sum of its digits<br /><br />90 is the smallest number having 6 representations as a sum of four positive squares<br /><br />90 is the number of degrees in a right angle. Moreover, as a compass direction, 90 degrees corresponds to east. Which reminds me of a fun math joke:"The number you have dialed is imaginary. Please rotate you phone by 90 degrees and dial again."<br /><br />And 90 is the sum of the first 9 consecutive even numbers, the sum of consecutive integers in two different ways, the sum of two consecutive primes, and of six consecutive primes(in two distinct ways), and the sum of five consecutive squares. (all proofs left to the reader.)<br /><br />(90<sup>3</sup> - 1)/(90 - 1) is a Mersenne prime. *Prime Curios The bases in Major League Baseball are 90 feet apart. <br /><hr /></div></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4490843217324699740.post-24186338696199615822020-01-27T18:48:00.038-08:002021-04-05T15:00:41.981-07:00Number Facts for Every Year Day ((31-60)) from "On This Day in Math"<b>The 31st day of the year</b>; 31 = 2<sup>2</sup> + 3<sup>3</sup>, i.e., The eleventh prime, and third Mersenne prime, it is also the sum of the first two primes raised to themselves. *<a href="http://www.numbergossip.com/31" target="_blank">Number Gossip</a> (<i>Is there another prime which is the sum of consecutive primes raised to themselves? </i><br /><i> A note from Andy Pepperdine of Bath informed me that</i> \(2^2 + 3^3 +5^5 + 7^7 = 826699 \), a prime.<br /><br />The sum of the first eight digits of pi = 3+1+4+1+5+9+2+6 = 31. *Prime Curios <br /><br />There are only 31 numbers that cannot be expressed as the sum of distinct squares. *Prime curios<br /><br />31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge. <br /><br />The numbers 31, 331, 3331, 33331, 333331, 3333331, and 33333331 are all prime. For a time it was thought that every number of the form 3w1 would be prime. However, the next nine numbers of the sequence are composite *Wik<br /><br />31 = 5^0 + 5^1 + 5^2 and also 31 = 2^0 + 2^1 + 2^2 + 2^3 + 2^4. *Mario Livio says that there are only two known numbers that can be expressed as consecutive powers of a number in two different ways. The second is 8191, which can be expressed as consecutive powers of two, and of ninety.<br /><br />\(\pi^3\) (almost)=31 (31,006...)<br /><br />There are only 31 numbers that cannot be expressed as the sum of distinct squares.<br /><br />31 is the minimum number of moves to solve the Towers of Hanoi problem. The general solution for any number of discs is a Mersenne number of the form 2^n -1.<br /><br />Jim Wilder @wilderlab offered, The sum of digits of the 31st Fibonacci number (1346269) is 31.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-bpYsYq-Qssg/VtdtsDVDdhI/AAAAAAAAH8s/tTWYT6e_0Xg/s1600/31%2Bmph%2B%25281%2529.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://1.bp.blogspot.com/-bpYsYq-Qssg/VtdtsDVDdhI/AAAAAAAAH8s/tTWYT6e_0Xg/s320/31%2Bmph%2B%25281%2529.jpg" /></a></div>If you like unusual speed limits, the speed limit in downtown Trenton, a small city in northwestern Tennessee, is 31 miles per hour. And the little teapot on the sign? Well, Trenton also bills itself as the teapot capital of the nation. The 31 mph road sign seems to come from a conflict between Trenton and a neighboring town which I will not name ,...but I will tell you they think of themselves as the white squirrel capital. <br /><br />31 is also the smallest integer that can be written as the sum of four positive squares in two ways 1+1+4+25; 4+9+9+9.<br /><br /> 31 is an evil math teacher number. The sequence of the maximum number of regions obtained by joining n points around a circle by straight lines begins 2, 4, 8, 16... but for five points, it is 31.<br /><br />@JamesTanton posted a mathematical fact and query regarding 31. <span face=", "blinkmacsystemfont" , "segoe ui" , "roboto" , "ubuntu" , "helvetica neue" , sans-serif" style="background-color: white; color: #38444d; font-size: 14px; white-space: pre-wrap;">31 =111(base 5) =11111(base 2) and 8191 =111(base 90) = 111111111111(base 2) are the only two integers known to be repunits at least 3 digits long in two different bases. </span><span face=", "blinkmacsystemfont" , "segoe ui" , "roboto" , "ubuntu" , "helvetica neue" , sans-serif" style="background-color: white; color: #38444d; font-size: 14px; white-space: pre-wrap;">Is there an integer with representations 10101010..., ,at least three digits, in each of two different bases? </span><span face=", "blinkmacsystemfont" , "segoe ui" , "roboto" , "ubuntu" , "helvetica neue" , sans-serif" style="background-color: white; color: #38444d; font-size: 14px; white-space: pre-wrap;"></span> <span face=", "blinkmacsystemfont" , "segoe ui" , "roboto" , "ubuntu" , "helvetica neue" , sans-serif" style="background-color: white; color: #38444d; font-size: 14px; white-space: pre-wrap;">Which made me wonder, are there other pairs that are repdigits (all alike, but not all units) in two (or more) different bases? </span><br /><hr /><b>The 32nd day of the year; </b> 32 is conjectured to be the highest power of two with all prime digits. *Number Gossip (Could 27 hold the similar property for powers of three?)<br /><br /> Also, 131 is the 32nd prime and the sum of the digits of both numbers is 5. 32 & 131 is the smallest n, P(n) with this property. \( 32 = 1^1 + 2^2 + 3^3 \) <br /><br />A fermat prime is a prime number of the form \(2^{2^n} +1 \) and five are known (3, 5, 17, 257, 65537). Their product is \( 2^{32} -1\)<br /><br />32! - 1 and 33!-1 are both primes<br />[David Marain pointed out that the products of the first n are all expressible in 2<sup>n</sup>-1 form, \( 3x5 = 2^4-1, 3x5x17 = 2^8-1\), and \(3x5x7x257 = 2^{16}-1 \) ]<br /><br /> On an 8x8 chessboard, the longest closed non-crossing knight's path is 32 moves.<div><br /></div><div><span style="background-color: rgba(0, 0, 0, 0.03); color: #0f1419; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 15px; white-space: pre-wrap;">The integers 1 through n=32 can be arranged in a circle so that every adjacent pair sums to a perfect square. (Try it!) Can those numbers be arranged in a circle such that any THREE adjacent numbers sum to a perfect square? *HT Matt Enlow</span><br /><hr /><b>The 33rd Day of the Year</b>;<br />among the infinity of integers, there are only six that can not be formed by the addition of distinct triangular numbers. The largest of these is 33. <i>What are the other five?</i><br /><br /><i></i>33 = 1!+2!+3!+4! *jim wilder @wilderlab<br /><br />33 is the smallest n such that n, n+1 and n+2 are all semi-primes, the products of two primes. *Bob S McDonald<br /><br />32! - 1 and 33!-1 are both primes<br /><br /> The 33 letter Dutch word nepparterrestaalplaatserretrappen is the longest palindrome I know in any language. It means fake stairways from the ground floor to the sun lounge, made of steel plate. The shorter word "saippuakauppias" for a soap vendor is the longest single word palindrome in the world that is in everyday use. *Wiktionary<br /><br /> 10<sup>33</sup> is the largest known power of ten that can be expressed as the power of two factors neither of which contains a zero. 10<sup>33</sup> = 2<sup>33</sup> 5<sup>33</sup> = 8,589,934,592 x 116,415,321,826,934,814,453,125 *Cliff Pickover @pickover <br /><br />The smallest odd number n such that n+x! is not a prime, for any number x. <br /><br />33 is the smallest teo-digit palindrome in base ten which is also a palindrome in a smaller base.<br /><br /><hr /><b>The 3</b><b>4th day of the year</b>; 34 is the smallest integer such that it and both its neighbors are the product of the same number of primes.<br /><br /> 34 is the smallest number which can be expressed as the sum of two primes in four ways.*Prime Curios<br /><br /> A 4x4 magic square using the integers 1 to 16 has a magic constant of 34. An early example is in the tenth century Parshvanath Jain temple in Khajuraho. The image below was taken by Debra Gross Aczel, the wife of the late Amir D. Aczel who used the image in his last book, Finding Zero.<br />4x4 magic squares were <span class="st">written about in India by a mathematician named<i> </i>Nagarjuna as early as the first century. </span><span class="st"><i></i></span> <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-8GrQsAI0LUQ/VogTTnmG1YI/AAAAAAAAHWc/ol05xpijkJI/s1600/KhajurahoNumbers.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://1.bp.blogspot.com/-8GrQsAI0LUQ/VogTTnmG1YI/AAAAAAAAHWc/ol05xpijkJI/s200/KhajurahoNumbers.jpg" width="200" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-wGd4XQBG0qM/VogSvG-TpFI/AAAAAAAAHWU/lc9p47Uh-kc/s1600/4x4%2B%2Bmagic%2Bsquare%2Bindia.jpe" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-wGd4XQBG0qM/VogSvG-TpFI/AAAAAAAAHWU/lc9p47Uh-kc/s320/4x4%2B%2Bmagic%2Bsquare%2Bindia.jpe" /> </a></div><div class="separator" style="clear: both; text-align: center;"></div><hr /><b>The 35th Day of The Year, </b>There are 35 hexominos, the polyominoes made from 6 squares. *Number Gossip (I only recently learned that, Although a complete set of 35 hexominoes has a total of 210 squares, which offers several possible rectangular configurations, it is not possible to pack the hexominoies into a rectangle.) <br /><div class="separator" style="clear: both; text-align: center;"><b><a href="http://4.bp.blogspot.com/-aTwmb7lBWMw/VrKX_EH7ieI/AAAAAAAAHxA/E2XIvZLepOY/s1600/hexominoes.jpe" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="284" src="https://4.bp.blogspot.com/-aTwmb7lBWMw/VrKX_EH7ieI/AAAAAAAAHxA/E2XIvZLepOY/s320/hexominoes.jpe" width="320" /></a></b></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div>The longest open uncrossed (doesn't cross it's own path) knight's path on an 8x8 chessboard is 35 moves. (longest cycle(end where you start) is only 32 moves) <br /><div class="separator" style="clear: both; text-align: center;"><b><a href="http://1.bp.blogspot.com/-ef1dr-pyHLQ/VrKYdF4IkZI/AAAAAAAAHxE/j5k-Fg9Jz5E/s1600/UncrossedKnightsPath8x8.svg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-ef1dr-pyHLQ/VrKYdF4IkZI/AAAAAAAAHxE/j5k-Fg9Jz5E/s1600/UncrossedKnightsPath8x8.svg" /></a></b></div><b></b><br /><div class="separator" style="clear: both; text-align: center;"><b><br /></b></div>In Base 35 (A=10, B=11, etc) NERD is Prime, \(23*35^3+14*35^2+27*35+13 = 1,004,233 \).<div>Chaw wrote, "<span face="Roboto, RobotoDraft, Helvetica, Arial, sans-serif" style="background-color: white; color: #3c4043; font-size: 16px;">Re. the observation that "NERD" is prime in base 35: I think base 36 is a lot more natural than base 35, given the conventional 10 digits and 26 Latin letters, which makes the following more interesting: NERDIEST is prime in base 36." </span><br /><hr /><div><b>The 36th Day of the Year</b>, The 36th day of the year; 36 is the smallest non trivial number which is both triangular and square. It's also the largest day number of the year which is both. What's the next? You can find an infinity of them using this beautiful formula from Euler, Hat Tip to Vincent PANTALONI @panlepan</div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/--HjLxd0DpDI/WgYkSK6F4sI/AAAAAAAAI2Q/d7p-JO9zVhQP5P30ye6hxL1bVq4_DYYTgCLcBGAs/s1600/square%2526triangulareuler.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="326" data-original-width="1024" height="127" src="https://2.bp.blogspot.com/--HjLxd0DpDI/WgYkSK6F4sI/AAAAAAAAI2Q/d7p-JO9zVhQP5P30ye6hxL1bVq4_DYYTgCLcBGAs/s400/square%2526triangulareuler.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div>36 is the sum of the first three cubes, \(1 ^3 + 2^3 + 3^3 = 36\) The sums of the first n cubes is always a square number. \(\sum_{k=1}^n k^3 = (\frac{(n)(n+1)}{2}) ^2\) <i>Note that this sequence and its formula were known to (and possibly discovered by) Nicomachus, 100 CE</i>) <br /><br />The sum of the first 36 integers, \(\sum_{k=1}^{36} k = 666\) the so called "number of the beast." Notice that a \(6^2\) (a triangular number) consecutive integers forms a repdigit triangular number, 666. <br /><br />36 itself is the last year day which is both a square and a triangular number. The next square that is a triangular number is 1225. (The square of 36-1)</div><div>36 is also the smallest triangular number that is the sum of two consecutive triangular numbers</div><div> And Mario Livio pointed out in a tweet that Feb 5 is 5/2 in European style dating, and 52 is the maximum number of moves needed to solve the "15" sliding puzzle from any solvable position.<br /><br /> The Kiwi's seeds divide the circle into 36 equal sections. Nature's protractor. *Matemolivares@Matemolivares <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-wws66b_FakI/V9McuGPTTiI/AAAAAAAAIdA/WihOdfNizM4Ik7i3OGCUrQY0e_HXQaTjQCLcB/s1600/Kiwi%2Bprotractor.jpe" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="318" src="https://3.bp.blogspot.com/-wws66b_FakI/V9McuGPTTiI/AAAAAAAAIdA/WihOdfNizM4Ik7i3OGCUrQY0e_HXQaTjQCLcB/s320/Kiwi%2Bprotractor.jpe" width="320" /></a></div><br />A special historical tribute to 36: The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler in 1782. He asked if it were possible to place officers of six ranks from each of six regiments in a 6x6 square so that no row or column would have an officer of the same rank, or the same regiment. Euler suspected that it could not be done. Euler knew how to construct such squares for nxn when n was odd, or a multiple of four, and he believed that all such squares with n = 4m+2 (6, 10, 14...) were impossible ( Euler didn't say it couldn't be done. He just said that his method does not work for numbers of that form.) Proof that he was right for n=6 took a while. French mathematician (and obviously a very patient man) Gaston Tarry proved it in 1901 by the method of exhaustion. He wrote out each of the 9408 6x6 squares and found that none of them worked. Then in 1959, R.C. Bose and S. S. Shrikhande proved that all the others could be constructed. So the thirty-six square is the <i>only one that can't be done.</i><br /><br />Jim Wilder sent \(36^2 = 1296\) and 1 + 29 + 6 = 36 <br /><i><br /></i>Chaw wrote, "<span face="Roboto, RobotoDraft, Helvetica, Arial, sans-serif" style="background-color: white; color: #3c4043; font-size: 16px;">Re. the observation that "NERD" is prime in base 35: I think base 36 is a lot more natural than base 35, given the conventional 10 digits and 26 Latin letters, which makes the following more interesting: NERDIEST is prime in base 36." </span><i><br /></i><br /><br /> Touchard (1953) proved that an <b>odd</b> perfect number, if it exists, must be of the form 12k+1 or 36k+9 *Wolfram Mathworld</div><div><br /></div><div>36^4 = <span data-sheets-formula="=R[0]C[-4]^4" data-sheets-value="{"1":3,"3":1679616}" style="font-family: Arial; font-size: 10pt; text-align: right;">1679616, and the sum of the digits is 36. It is the largest number for which the sum of the digits of n^4 is equal to n .There are three smaller numbers (Not counting 1) which have this property also.</span></div><div><span data-sheets-formula="=R[0]C[-4]^4" data-sheets-value="{"1":3,"3":1679616}" style="font-family: Arial; font-size: 10pt; text-align: right;">and 36^5 = </span><span style="font-family: Arial; font-size: 10pt; text-align: right;">60466176 which <b>also </b>has a sum of digits of 36. I haven't found any other numbers where sum of digits of N^a and N^b are the same. </span></div><div><hr b="" /><b>The 37th Day of the Year</b>.</div><div>The 37th day of the year; 37 is the only prime with a three digit period for the decimal expansion of its reciprocal, 1/37 = .027027.... But 37 has a strange affinity with 27, which also has a three digit period for its reciprocal, .037037..., The affinity, of course, is due to 27 x 37 = 999</div><div>Can we call numbers like this, <b>amicable reciprocals</b>?<br />and Alex Kontorovich found 1/1287 = .000777000777... and yeah, 1/777 = .001287001287... Now you can find some of your own (and make sure to send me a note)!<br /><br />Speaking of prime periods, students should know that the longest repeating decimal for the inverse of a prime number p is p-1. It seems that about 37% of the primes reach this max, but not 37 as mentioned above. </div><div><br /> Big Prime::: n = integer whose digits are (left to right) 6424 copies of 37, followed by units digit of 3, is prime (n = 3737...373 has 12849 digits) *Republic of Math<br /><br />An amazing reversal: 37 is the 12th prime; and 73 is the 21st prime . This enigma is the only known combination.<br /><br />37 is the last year day such that the sum of the squares of the first n primes, is divisible by n. There are only three such numbers in the days of the year. Two of them are primes themselves.<br /><br />If you use multiplication and division operations to combine Fibonacci numbers, (for example, 4 = 2^2, 6 = 2·3, 7 = 21/ 3 ,...) you can make almost any other number. Almost, but you can't make 37. In fact, there are 12 numbers less than 100 that can not be expressed as "Fibonacci Integers" *Carl Pomerance, et<br /><br />37! + 1 is a prime. It is the sixth Year Day for which this is true, and the last prime year day. There are only 13 Year Days for which n! + 1 is prime.<br /><br />To represent every integer as a sum of fifth powers requires at most 37 integers.<br /><br />The last odd Roman numeral alphabetically is XXXVII (37). *prime Curios <br /><br />Can we call numbers like this, <b>amicable reciprocals</b>?<br />and Alex Kontorovich found 1/1287 = .000777000777... and yeah, 1/777 = .001287001287...<br /><hr /><b>The 38th Day Of the Year , </b>31415926535897932384626433832795028841 is a prime number. BUT, It’s also the first 38 digits of pi.<br /><br /> 38 is the largest even number so that every partition of it into two odd integers must contain a prime.<br /><br />38 is the largest even number that can only be expressed as the sum of two distinct primes in one way. (31 + 7)<br /><br /> 38 is the sum of squares of the first three primes \(2^2 + 3^2 + 5^2 = 38 \). *Prime Curios<br /><br /><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;">Although we've had some unusual shaped flags, usually the star field is in a rectangle with the stars displaying some kind of (generally rectangular) similarity. Some have strayed greatly from the rectangle form however. This one with 38 stars from 1877 until 1890 is an example.</span></div><div><span face="Arial, Tahoma, Helvetica, FreeSans, sans-serif" style="background-color: white; color: #222222;"><br /></span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-xTrt7ol2HO8/YB676wtXdsI/AAAAAAAANW8/JZ_yQoucI1okgjgpw08025B_Bn5UvRQDQCLcBGAsYHQ/s322/38%2Bstar%2Bflag.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="229" data-original-width="322" src="https://1.bp.blogspot.com/-xTrt7ol2HO8/YB676wtXdsI/AAAAAAAANW8/JZ_yQoucI1okgjgpw08025B_Bn5UvRQDQCLcBGAsYHQ/s320/38%2Bstar%2Bflag.jpg" width="320" /></a></div><br /></div><div> At the beginning of the 21st Century there were 38 known Mersenne Primes. As of this writing, there are 51, the last being discovered in Dec of 2018..<br /><br /> 38 is also the magic constant in the only possible magic Yhexagon which utilizes all the natural integers up to and including 19. It was discovered independently by Ernst von Haselberg in 1887, W. Radcliffe in 1895, and several others. Eventually it was also discovered by Clifford W. Adams, who worked on the problem from 1910 to 1957. He worked on the problem throughout his career as a freight-handler and clerk for the Reading Rail Road by trial and error and after many years arrived at the solution which he transmitted to Martin Gardner in 1963. Gardner sent Adams' magic hexagon to Charles W. Trigg, who by mathematical analysis found that it was unique disregarding rotations and reflections. <br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-9WAmozF7GDo/TyQwBOlf7LI/AAAAAAAAD8I/B1cUlvqgJpc/s1600/MagicHexagon-Order3-a.svg" style="margin-left: auto; margin-right: auto;"><img border="0" height="185" src="https://2.bp.blogspot.com/-9WAmozF7GDo/TyQwBOlf7LI/AAAAAAAAD8I/B1cUlvqgJpc/s200/MagicHexagon-Order3-a.svg" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Wik</td></tr></tbody></table><br /><br /><hr /><b>The 39th Day of the Year,</b> 39 is the smallest number with multiplicative persistence 3. [<i>Multiplicative persistence is the number of times the digits must be multiplied until they produce a one digit number; 3(9)= 27; 2(7) = 14; 1(4)=4. Students might try to find the smallest number with multiplicative persistence of four, or prove that no number has multiplicative persistence greater than 11</i>]<br /><br />39 = 3¹ + 3² + 3³ *jim wilder @wilderlab An Armstrong (or <i>Pluperfect digital invariant</i>) number is a number that is the sum of its own digits each raised to the power of the number of digits. For example, 371 is an Armstrong number since \(3^3+7^3+1^3 = 371\). The largest Armstrong number in decimal numbers has 39 digits. (115,132,219,018,763,992,565,095,597,973,522,401 is the largest) (Armstrong numbers are named for Michael F. Armstrong who named them for himself as part of an assignment to his class in Fortran Programming at the University of Rochester \)<br /><br /> I find it interesting that 39 = 3*13, and is the sum of all the primes from 3 to 13, 39=3+5+7+11+13, these are sometimes call ed straddled numbers.<br /><br />39 is the smallest positive integer which cannot be formed from the first four primes (used once each), using only the simple operations +, -, *, / and ^. Prime Curios.<br /><br />The number formed by concatenating the non-prime integers 1 through 39 is the smallest such prime: 1468910121415161820212224252627283032333435363839. Prime Curios.<br /><br />\$ 3^{39} = 4052555153018976267 \$ is the smallest power of three which is pandigital, with all ten decimal digits. The number is 19 digits long. *@Fermat's Librarty<br /><br /><br /><br /><hr /><b>The 40th Day of the Year</b>: in English forty is the only number whose letters are in alphabetical order.<br /><br /> There are 40 solutions on for the 7 queens problem. placing seven chess queens<a href="http://en.wikipedia.org/wiki/Queen_%28chess%29" title="Queen (chess)"></a> on a 7x7 chessboard so that no two queens threaten each other.<br /><br /> -40 is the temperature at which the Fahrenheit and Celsius scales correspond; that is, −40 °F = −40 °C. <br /><br />Euler first noticed (in 1772) that the quadratic polynomial P(n) = n<sup>2</sup> + n + 41 is prime for all non-negative numbers less than 40.<br /><br />Paul Halcke noted in 1719 that the product of the aliquot parts of 40 is equal to 40 cubed. 1*2*4*5*8*10*20 = 64000 = 40<sup>3</sup>. He found the same is true for 24.<br /><br /> And.... forty is the highest number ever counted to on Sesame Street. <br /><br />40 = 2^3+5, the first three primes in order.<br /><br /><hr /><b>The 41st Day of the Year</b>:<br />Euler (1772) observed that the polynomial f(x)= x2 + x + 41 will produce a prime for any integer value of x in the interval 0 to 39.<br /><br /> In 1778 Legendre realized that x2 - x + 41 will give the same primes for interval (1-40). n^2 + n + 41 is prime for n = 0 ... 39 and Is prime for nearly half the values of n up to 10,000,000. *John D. Cook<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">The smallest prime whose cube can be written as sum of three cubes in two ways (41</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">3</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> = 40</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">3</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> + 17</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">3</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> + 2</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">3</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> = 33</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">3</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> + 32</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">3</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> + 6</span><sup style="background-color: #fcfcfc; font-family: "Comic Sans MS", Georgia, sans-serif;">3</sup><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">). *Prime Curios</span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span> If you multiply 41 by any three digit number to produce a five digit number, every cyclic representation of that number formed by moving the last digit to the front is also divisible by 41. (for example 41*378 = 15,498. 41 will also divide 81,549; 98154; 49815; and 54,981 *The Moscow Puzzles<br /><br /> 41 can be expressed as the sum of consecutive primes in two ways, (2 + 3 + 5 + 7 + 11 + 13), and the (11 + 13 + 17).<br /><br /> The sum of the digits of 41 (5) is the period length of its reciprocal, 1/41 =.0243902439,,, It is the smallest number with a period length of five.<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">41 is the largest known prime formed by the sum of the first Mersenne primes in logical order (3 + 7 + 31) *Prime Curios</span><br /><div><br />Incredibly, if you take any two integers that sum to 41, a+b =41, then a^2 + b is a prime, for example, 20^2 + 21 = 421<br /><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">Starting with 41, if you add 2, then 4, then 6, then 8, etc., you will have a string of 40 straight prime numbers. *Prime Curios</span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;">The 41st </span>Mersenne<span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"> to be found = 2^24036583-1. *Prime Curios</span><br /><span face=""comic sans ms" , "georgia" , sans-serif" style="background-color: #fcfcfc; font-size: 16px;"><br /></span></div> And even more from @Math Year-Round 41=1!+2!+3!+1¹+2²+3³ <br /><hr /><b>The 42nd Day of the Year: </b><br />in The Hitchhiker's Guide to the Galaxy, the Answer to the Ultimate Question of Life, The Universe, and Everything is 42. The supercomputer, Deep Thought, specially built for this purpose takes 7½ million years to compute and check the answer. The Ultimate Question itself is unknown. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Qe70JPoDiQk/XFoGv-i--_I/AAAAAAAAJSA/zSgz8ORaC-8zILaebH3dib3aCA6Hh3pNgCLcBGAs/s1600/42%2Bwolfram%2Balpha.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="142" data-original-width="541" height="103" src="https://2.bp.blogspot.com/-Qe70JPoDiQk/XFoGv-i--_I/AAAAAAAAJSA/zSgz8ORaC-8zILaebH3dib3aCA6Hh3pNgCLcBGAs/s400/42%2Bwolfram%2Balpha.jpg" width="400" /></a></div><br />There is only one scalene triangle in simplest terms with integer sides and integer area of 42, it's perimeter is also 42. (There are only three integer (non-right) triangles possible with area and perimeter equal and all integer sides.)<br /><br /> 42 is between a pair of twin primes (41,43) and its concatenation with either of them (4241, 4243) is also a prime, which means that 4242 is also between twin primes.<br /><br /> On September 6, 2019, Andrew Booker, University of Bristol, and Andrew Sutherland, Massachusetts Institute of Technology, found a sum of three cubes for \(42= (–80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 \). This leaves 114 as the lowest unsolved case. At the beginning of 2019, 33 was the lowest unsolved case, but Booker solved that one earlier in 2019. <br />42 is the largest number n such that there exist positive integers p, q, r with 1 = 1 / n + 1 / p + 1 / q + 1 / r <b><br /></b><br /><span color="rgba(0, 0, 0, 0.9)" face="-apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", "Fira Sans", Ubuntu, Oxygen, "Oxygen Sans", Cantarell, "Droid Sans", "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Lucida Grande", Helvetica, Arial, sans-serif" style="background-color: white; font-size: 14px;"> In 1954, researchers at the University of Cambridge looked for solutions of the equation x^3 + y^3 + z^3 = k, with k being all the numbers from 1 to 100. As of late 2019, all numbers had been solved except 42, which proved to be especially challenging. That is until University of Bristol’s Professor Andrew Booker and MIT Professor Andrew Sutherland solved the equation with the help of @CharityEngine, a crowdsourcing platform that harnesses idle, unused computing power from more than 500,000 home PCs. *Mehmet Aslan</span></div><div><br /><div class="entity-hovercard__info ml2" style="background: rgb(255, 255, 255); border: 0px; box-sizing: inherit; color: rgba(0, 0, 0, 0.9); flex: 1 1 auto; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", "Fira Sans", Ubuntu, Oxygen, "Oxygen Sans", Cantarell, "Droid Sans", "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Lucida Grande", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0px; margin-left: 8px !important; margin-right: 0px; margin-top: 0px; margin: 0px 0px 0px 8px; min-width: 0px; overflow-wrap: break-word; padding: 0px; vertical-align: baseline;"><h1 class="entity-hovercard__title-container t-14 t-black t-bold" style="background: transparent; border: 0px; box-sizing: inherit; color: var(--color-text); font-size: 1.4rem; line-height: 1.42857; margin: 0px; outline: 0px; padding: 0px; position: relative; vertical-align: baseline;"><span class="t-14 t-black--light t-normal entity-hovercard__distance-badge distance-badge t-black--light t-14 separator ember-view" color="var(--color-text-low-emphasis)" data-test-distance-badge="" id="ember777" style="background: transparent; border: 0px; box-sizing: inherit; font-size: 1.4rem; font-weight: 400; line-height: 1.42857; margin: 0px; outline: 0px; padding: 0px; position: relative; vertical-align: baseline;"><span class="visually-hidden" style="background: transparent; border: 0px; box-sizing: inherit; clip: rect(1px, 1px, 1px, 1px); display: block; font-size: 14px; height: 1px; margin: -1px; outline: 0px; overflow: hidden; padding: 0px; position: absolute; vertical-align: baseline; white-space: nowrap; width: 1px;">out of network</span></span></h1></div>Given 27 same size whose nominal values progress from 1 to 27, a 3 × 3 × 3 magic cube can be constructed such that every row, column, and corridor, and every diagonal passing through the center, is composed of 3 numbers whose sum of values is 42.<br /><hr /><b>The 43rd Day of the Year.:</b><br /><b> T</b>he McNuggets version of the coin problem was introduced by Henri Picciotto, who included it in his algebra textbook co-authored with Anita Wah.Picciotto thought of the application in the 1980s while dining with his son at McDonald’s, working the problem out on a napkin. A McNugget number is the total number of McDonald’s Chicken McNuggets in any number of boxes.The original boxes (prior to the introduction of the Happy Meal-sized nuggetboxes) were of 6, 9, and 20 nuggets.According to Schur’s theorem, since 6, 9, and 20 are relatively prime,any suﬃciently large integer can be expressed as a linear combination of these three. Therefore, there exists a largest non-McNugget number, and all integers larger than it are McNugget numbers.That number is 43, so how many of each size box gives the McNugget number 44?<br /><br /><br />43 is the number of seven-ominoids. (shapes made with seven equilateral triangles sharing a common edge.)<br /><br /> In March of 1950, Claude Shannon calculated that there are appx \( \frac{64!}{32!} (8!)2(2!)6 \), or roughly 10<sup>43</sup> possible positions in a chess match.<br /><br /> Planck time (~ 10<sup>-43</sup> seconds) is the smallest measurement of time within the framework of classical mechanics. That means that if you could make one unique chess position in each Planck time, you could run through them all in one second.<br /><br /> What is the minimum number of guests that must be invited to a party so that there are either five mutual acquaintances, or five that are mutual strangers? (Sorry we still don't know :-{ But the smallest number must be 43 or larger). I think that means that for any number of points on a circle less than 43, if you colored every segment connecting two of them either red or black, there would be no complete graph of five vertices (K(5)) with all edges of the same color. [And there are 43 choose 5 or 962,598 possible choices of complete graphs to choose from.]<br /><br /> According to Benford's Law, the odds that a random prime begins with a prime digit is more than 43%<br /><br />Every solvable configuration of the Fifteen puzzle can be solved in no more than 43 multi-tile moves (i.e. when moving two or three tiles at once is counted as one move) <br /><br />43/100 or more exactly 3/7 shows up in a good approximation for the area of an equilateral triangle: "Gerbert of Aurillac (later Pope Sylvester II) referred to the equilateral triangle as “mother of all figures” and provided the formula A ≈ s^ 2 · 3/7 which estimates its area in terms of the length of its side to within about 1.003% ( N. M. Brown, The Abacus and the Cross, Basic, 2010, p. 109). 3/7 = .428571… sqrt(3)/4 = .433012<br /><br />43 is the smallest (non-trivial) number that is equal to the consecutive powers of its digits, \(4^2 + 3^3 = 43\) . There are two more two digit numbers that fit this pattern. There are also two three digit year dates that fit if you restrict the powers to consecutive integers starting with 1. </div><div> <br />And if 42 was the meaning of life, the universe, and everything, just imagine that 43 is MORE than that! </div><div><br /></div><div>Jim Wilder shared \(43^7= 271818611107\) which has a digit sum of 43. Wonder how often you can find a multidigit number n so that n^k (for some k=2 through 9 will have a digit sum of n, or a digit sum which is m*n for some integer m? 27^3 = 19683 and 27^7 = 10460353203 came to mind. Rare, or not rare? When I checked 26^3, it also worked but not to seventh power. 53^7 = <img alt="1174711139837" class="_3vyrn" src="https://www4b.wolframalpha.com/Calculate/MSP/MSP74823e62di5fee1e6h3000038g5bg099gh3i4cf?MSPStoreType=image/gif&s=53" style="background-color: white; border: 0px; box-sizing: border-box; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; max-width: 100%; outline: none; padding: 0px; vertical-align: baseline; width: 113px;" /> also works, it seems </div><div><hr /><b>The 44th Day of the Year: </b><br />there are 44 ways to reorder the numbers 1 through five so that none of the digits is in its natural place. This is called a derangement. The number of derangements of n items is an interesting study for students. Some <a href="http://pballew.blogspot.com/2009/05/on-trail-of-subfactorial-notation.html" target="_blank">historical notes from here</a>. If you had five letters for five different people and five envelopes addressed to the five people, there are 44 ways to put every letter in the wrong envelope.<br /><br />44 is the sum of the first emirp (prime which is prime with digits reversed) pair, 13 and 31.*Prime Curios<br /><br />44 is the smallest number such that it and the next number are the product of a prime and another distinct prime squared (44 = 2<sup>2</sup>*11 and 45 = 3<sup>2</sup>*5).<br /><br />All even perfect numbers greater than 6 end in 44 in base six, as do all powers of ten greater than 10. *Lord Karl Voldevive @Karl4MarioMugan (students should be encouraged to understand that the converse of these statements is not true by finding exceptions.)<br /><br />44 and 45 form the first pair of consecutive numbers that are the product of a prime and the square of a prime. 44 = 2^2 * 11 and 45 = 3^2 * 5<br /><br />44 is a palindrome in base ten, but not in any smaller base. Only three of the ninr two-digit palindromes in base ten are palindromes in any smaller base. Find them!<br /><br />An Euler brick, named after Leonhard Euler, is a cuboid whose edges andface diagonals all have integer lengths. A primitive Euler brick is an Eulerbrick whose edge lengths are relatively prime.The smallest Euler brick, discovered by Paul Halcke in 1719, has edges( a,b,c ) = (44 , 117 , 240) and face diagonals 125, 244, and 267. <br /><hr /><b>The 45th Day of the Year: </b><br />45 is the third Kaprekar number. (45<sup>2</sup> = 2025 and 20 + 25 = 45) The next two Kaprekar numbers both have two digits, can you find them? More unusual, it is also a Kaprekar number with third powers, 45^3 = 91125 and 9 + 11 + 25 = 45. But Wait! There's more. 45^4 = 4,100,625 and yes, 4 + 10 + 06 + 25 = 45. There is no other number known that is a Kaprekar number in all three powers.<br /><br />45 is the 9th triangular number, the sum of the digits from 1 through 9.<br /><br />45-2<sup>n</sup> for n=1 through 5 forms a prime<br /> I found these on a post at the <a href="http://www.futilitycloset.com/" target="_blank">Futility Closet</a> by Greg Ross: 45<sup>2</sup> = 2025 20 + 25 = 45 45<sup>3</sup> = 91125 9 + 11 + 25 = 45 45<sup>4</sup> = 4100625 4 + 10 + 06 + 25 = 45 <br /><br />45 is a palindrome in base 2 {101101} and base 8{55}<br /><br />44 and 45 form the first pair of consecutive numbers that are the product of a prime and the square of a prime. 44 = 2^2 * 11 and 45 = 3^2 * 5<br /> And a paradoxical anagram about 45; Over fifty = forty-five <b><br /></b><br /><br />The 45th row of Pascal's Arithmetic Triangle has 30 even numbers, the 60th row, has 45 even numbers. 45 is the smallest odd number n that has more divisors than n+1 and that has a larger sum of divisors than n+1 <br /><br />The 45th parallel, halfway between the North Pole and Equater, runs just outside my family home in Elk Rapids, Michigan.<br /><hr /><b>The 46th Day of the Year:</b><br /><a href="http://2.bp.blogspot.com/-xvjPfBaj0c0/VPkfTUVOpBI/AAAAAAAAGfE/Z-HYjPJ1Oe0/s1600/46th%2BMersenne%2Bprime%2Bin%2BTime%2Bmag.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://2.bp.blogspot.com/-xvjPfBaj0c0/VPkfTUVOpBI/AAAAAAAAGfE/Z-HYjPJ1Oe0/s320/46th%2BMersenne%2Bprime%2Bin%2BTime%2Bmag.jpg" /></a></div> there are 46 fundamental ways to arrange nine queens on a 9x9 chessboard so that no queen is attacking any other. (Can you find solutions for smaller boards?)<br /><br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">46 is the largest even integer that cannot be expressed as a sum of two </span>abundant numbers<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">.</span><br /><span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;"><br /><span style="color: black; font-size: medium;">46 can be expressed as the sum of primes using the first four natural numbers once each, </span><span style="color: black; font-size: medium;">46 = 41 + 3 + 2</span><span style="color: black; font-size: medium;">41 + 23 = 64.</span><div style="color: black; font-size: medium;">It can also be done to its reversal 41 + 23 = 64.</div><div style="color: black; font-size: medium;"><br /></div></span>46 is the ninth "Lazy Caterer" number. The maximum number of pieces that can be formed with 9 straight cuts across a pancake.<br /><b 46="" 4="" 5="" a="" and="" base="" br="" in="" is="" nbsp="" palindrome=""><br /></b>46 is the number of integer partitions of 18 into distinct parts.<br /><br />46 is a palindrome in both base 4 and base 5<br /><br />On Oct 29, 2008 the 46th discovered Mersenne Prime, then the world's largest prime was featured in Time magazine as one of the "great inventions" of the year. It was discovered by Smith, Woltman, Kurowski, et al. of the GIMPS (Great Internet Mersenne Prime Search) program.<br /> Three more have been discovered since, one of which is smaller than this one, so while it was 46th discovered, it is 47th in rank. . <br /><br />46^5 = 205962976, with a digit sum of 46. 46^8 = 20047612231936 also with a digit sum of 46. 46 is the second smallest number which has two expressions of n^k = digit sum of n.<br /><hr /><b>The 47th Day of the Year:</b><br />47 is a Thabit number, named after the Iraqi mathematician Thâbit ibn Kurrah number, of the form 3 * 2<sup>n</sup> -1 (sometimes called 3-2-1 numbers). He studied their relationship to Amicable numbers. 47 is related to the amicable pair, (<span face="sans-serif" style="background-color: white; color: #222222; font-size: 14px;">17296, 18416)</span> All Thabit numbers expressed in binary end in 10 followed by n ones, 47 in binary is 101111. (The rule is that if p=3*2<sup>n-1</sup> -1, q= 3*2<sup>n</sup> -1, and r = 9*2<sup>n-1</sup> -1, are all prime, then 2<sup>n</sup>pq and 2<sup>n</sup>r are amicable numbers.<br /><br />3^3^3^3^3^3^3 has 47 distinct values depending on parentheses. *Math Year-Round @MathYearRound<br /><br />666<sup>47</sup> has a sum of digits equal to the Beast Number, 666 *Prime Curios<br /><br />47<sup>9</sup> can be written as the sum of distinct smaller 9th powers.*Prime Curios<br /><br />"The 47 Society is an international interest-group that follows the occurrence and recurrence of the quintessential random number: 47. Many suspect that the coincidental nature of 47 carries some mystical, metaphysical and/or scientific significance." *<a href="http://www.47.net/47society/%20" target="_blank">http://www.47.net/47society/ </a><br /><br /><br />Mario Livio has pointed out that this date written month day as 216, 216=6<sup>3</sup> and also 216=3<sup>3</sup>+4<sup>3</sup>+5<sup>3</sup><br /><br /> The 47th day gives me a reason to include this brief story of Thomas Hobbes from Aubrey's "Brief Lives". The 47th proposition of Libre I of The Elements (The Pythagorean Theorem) seemed so obviously false to him that, in following the reasoning back, his life was changed:<br /><br /><blockquote><b>He was (vide his life) 40 yeares old before he looked on geometry; which happened accidentally. Being in a gentleman’s library in . . . , Euclid’s Elements lay open, and ’twas the 47 El. libri I. He read the proposition. ‘By† G—,’ sayd he, ‘this is impossible!’So he reads the demonstration of it, which referred him back to such a propo- sition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps,(<i>and so back to the beginning</i>) that at last he was demonstratively convinced of that trueth. This made him in love with geometry. </b></blockquote><br /><hr /><b>The 48th Day of the Year: </b><br />48 is the smallest number with exactly ten divisors. <i>(This is an interesting sequence, and students might search for others. Finding the smallest number with twelve divisors will be easier than finding the one with eleven.)</i><br /><br /> 48 is also the smallest even number that can be expressed as a sum of two primes in 5 different ways: If n is greater than or equal to 48, then there exists a prime between n and 9n/8 This is an improvement on a conjecture known as Bertrand's Postulate. In spite of the name, many students remember it by the little rhyme, "Chebyshev said it, but I'll say it again; There's always a prime between n and 2n ." Mathematicians have lowered the 2n down to something like n+n<sup>.6</sup> for <i>sufficiently</i> large numbers.<br /><br /> 48 is the smallest betrothed (quasi-amicable) number. 48 and 75 are a betrothed pair since the sum of the proper divisors of 48 is 75+1 = 76 and the sum of the proper divisors of 75 is 48+1=49. (There is only a single other pair of betrothed numbers that can be a year day)<br /><br />And 48 x 48 = 2304 but 48 x 84 = 4032. (Others like this???)<br /><br />If you picked four prime numbers so that any collection of three of them had a prime sum, then the smallest sum you could get adding all four primes, is <b>48</b>. (5, 7, 17, 19). Can you find the next smallest?(suitable for middle school students to explore as there are many with modest size numbers)<br /><br />In 1719 Paul Halcke observed that the product of the aliquot divisors of 48 is equal to the fourth power of 48. 1*2*3*4*6*8*12*16*24= 5,308,416= 48<sup>4</sup>. 48 and 80 are the only two year dates for which this is true. <br /><br />48 is a Harshad Number from the Sanskrit for "joy-giver", since it is divisible by the sum of its digits. It is also one of the numbers cubed in the 11th Taxicab number <span color="rgba(0, 0, 0, 0.9)" face="-apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", "Fira Sans", Ubuntu, Oxygen, "Oxygen Sans", Cantarell, "Droid Sans", "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Lucida Grande", Helvetica, Arial, sans-serif" style="background-color: white; font-size: 14px;">110656 = 40 ^3 + 36 ^3 = 48 ^3 + 4 ^3. </span><br /><br />48 x 159 = 5346 And uses all nine digits <br /><hr /><b>The 49th Day of the Year:</b><br />lots of numbers are squareful (divisible by a square number) but 49 is the smallest number so that it, and both its neighbors are squareful. (<i>Many interesting questions arise for students.. what's next, can there be four in a row?, etc</i>)<br /><br /> And Prof. William D Banks of the University of Missouri has recently proved that every integer in base ten is the sum of 49 or less palindromes. (August 2015) (Building on Prof. Banks groundbreaking work, by February 22, 2016 JAVIER CILLERUELO AND FLORIAN LUCA had proved that for any base > 4 EVERY POSITIVE INTEGER IS A SUM OF THREE PALINDROMES )<br /><br /> The 49th Mersenne prime is discovered. On Jan 19th, 2016 The GIMPS program announced a new "largest known" prime, 2<sup>74,207,281</sup> -1. called M74,207,281 for short, the number has 22,338,618 digits. <br /><br />49 is the smallest square which is the sum of three consecutive primes.49= 17 + 19 + 23<br /><br />49 is the first square where the digits are squares, What's next?<div><br /></div><div>1, 25, 49 is the smallest arithmetic progression of three squares that I have ever found. 4, 100, and 196 come next . Is there one starting with nine? It is proven that an arithmetic progression of four squares in not possible. <br /><br />If you square 49, and take the sum of the digits of that square, you have 7, the square root of 49. How common is this?</div><div><br /></div><div>Student's are reminded that 8 x 6 + 1 is 49, an example of a beautiful mathematical truth that 8 x T + 1 is a square for any triangular number. <br /><br />1 / 49 = 0.0204081632 6530612244 8979591836 7346938775 51 and then repeats the same 42 digits. It's better than it looks. Write down all the powers of two, and then index them two to the right and add.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-4QeRql2vGO4/XkBhUX4LdeI/AAAAAAAALL8/odrKYxFK0xo27sNyVsQioXcaySFpVSWHgCLcBGAsYHQ/s1600/49%2Breciprocal.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="266" data-original-width="488" height="217" src="https://1.bp.blogspot.com/-4QeRql2vGO4/XkBhUX4LdeI/AAAAAAAALL8/odrKYxFK0xo27sNyVsQioXcaySFpVSWHgCLcBGAsYHQ/s400/49%2Breciprocal.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">*Wik</td></tr></tbody></table> <br />1, 25, 49 is the smallest arithmetic progression of three squares that I have ever found. 4, 100, and 196 come next . Is there one starting with nine? It is proven that an arithmetic progression of four squares in not possible. </div><div><br /></div><div><div>Everyone knows 25 is the hypotenuse of a Pythagorean right triangle with legs of 7 and 24. A Pythagorean triangle can never have two sides that are squares. When a square occurs as the shorter leg, and interesting pattern occurs:</div><div>leg leg hypotenuse</div><div>9 40 41</div><div>25 312 313</div><div>49 1200 1201 </div><div>81 3280 3281 </div><div><br /></div><div>good geometry students may already know that any (including all the squares above) odd number n larger than one, is the short leg of a right triangle with a difference of one between the other leg and the hypotenuse. The square of the odd leg is the sum of the other leg and hypotenuse</div><div> ODD Leg Even Leg Hypotenuse </div><div> 3 4 5 4 + 5 = 3^2</div><div> 5 12 13 12 + 13 = 5^2 </div><div> 7 24 25 24 + 25 = 7^2 </div><hr /><b>The 50th Day of the Year: </b><br />50 is the smallest number that can be written as the sum of two squares in two distinct ways 50 = 49 + 1 = 25 + 25. *Tanya Khovanova, <a href="http://www.numbergossip.com/" target="_blank">Number Gossip </a>(<i>What is the next, or what is the smallest number that can be written as the sum of two squares in three distinct ways?</i><br /><br />It is also the sum of three squares, 3^2 + 4^2 + 5^2 = 50 and of four squares, 1^2 + 2^2 + 3^2 + 6^2 = 50<br /><br />You can use the first nine consecutive primes to express 50 as the sum of primes in two different ways, :50 = 2 + 5 + 7 + 17 + 19 = 3 + 11 + 13 + 23.<br /><br />The number 50 is somewhat responsible for the area of number theory about partitions. In 1740 Philip Naudé the younger (1684-1747) wrote Euler from Berlin to ask “how many ways can the number 50 be written as a sum of seven different positive integers?” Euler would give the answer, 522, within a few days but would return to the problem of various types of partitions throughout the rest of his life.<br /><br />There is no solution to the equation φ(x) = 50, making 50 a nontotient (there is no integer, k, that has exactly 50 numbers below it that do not share a divisor with k, other than 1).<br /><br /><hr /><b> The 51st Day of the Year: </b><br />51 is the number of different paths from (0,0) to (6,0) made up of segments connecting lattice points that can only have slopes of 1, 0, or -1 but so that they never go below the x-axis. These are called <a href="http://mathworld.wolfram.com/MotzkinNumber.html" target="_blank">Motzkin Numbers</a>.<br /><br /> \(\pi(51) = 15\), the number of primes less than 51 is given by it's reversal, 15, and both numbers are products of Fermat Primes.<br /><br /> Jim Wilder pointed out that 51 is the smallest number that can be written as a sum of primes with the digits 1 to 5 each used once 2 + 3 + 5 + 41 = 51 (Students might explore similar problems using first n digits 2-9)<br /><br />51 can be expressed as the sum of four primes using only the digits from 1-5, 51 = 2 + 3 + 5 + 41.<br /><br />A triangle with sides 51, 52 and 53 has an integer area 1170 units<sup>2</sup>. These are called Heronian Triangles, or sometimes Super Heronian Triangles but I prefer to call them after the earliest study of them I have found, and will refer to them as Fleenor-Heronian triangles . (Guess I shouldn't be, but surprised how all of the triangles I could find with consecutive integer sides and integer area have final digits of 1,2,3 or 3,4,5) There are an infinite number of these with consecutive integers for sides. To find the even side, just take the expansion of \( (2 + \sqrt{3})^n \),and sum the rational terms, then double it to get the even side. The first three are 2, 7, giving us the even side of a 3,4,5 triangle and the 13, 14, 15 triangle. And if you expand \( (2 + \sqrt{3})^4 \) you get \( 8 + 12 \sqrt{3} + 3(2)3 + \sqrt{3^3} =26\) and we get the center side of the triangle above. (My thanks to @expert_says on twitter who sent me a link to two nice papers on this) (more notes about this in Day 52)<br /><br />And like any odd number, it is the sum of two consecutive numbers, 25+26 , and the difference of their squares \(26^2 - 25^2\)<br /><br /> And I just found this unusual reference, "Don’t be baffled if you see the number 51 cropping up in Chinese website names, since 51 sounds like 'without trouble' or 'carefree' in Chinese." at the <a href="http://www.archimedes-lab.org/" target="_blank">Archimedes Lab</a> <br /><br />Since 51 is the product of the distinct Fermat primes 3 and 17, a regular polygon with 51 sides is constructible with compass and straightedge, the angle π / 51 is constructible, and the number cos π / 51 is expressible in terms of square roots. <br /><br /><hr /><b>The 52nd Day of the Year</b>,<br />The month and day are simultaneously prime a total of 52 times in a non-leap year. *Tanya Khovanova, Number Gossip <i>How many times in a leap year ?</i><br /><br />52 is also the maximum number of moves needed to solve the 15 puzzle from the worst possible start<i>. *Mario Livio</i><br /><br /><div class="separator" style="clear: both; text-align: center;"><b 46="" 4="" 5="" a="" and="" base="" br="" in="" is="" nbsp="" palindrome=""><i><a href="https://2.bp.blogspot.com/-o0C7m15Zkd4/WKNFQyP-3kI/AAAAAAAAIp4/TPOXDamsUHkz_3eSC5XxL0NjS9bNANPqwCLcB/s1600/fifteen%2Bpuzzle.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-o0C7m15Zkd4/WKNFQyP-3kI/AAAAAAAAIp4/TPOXDamsUHkz_3eSC5XxL0NjS9bNANPqwCLcB/s1600/fifteen%2Bpuzzle.jpg" /></a></i></b></div><br />52 is the number of 8-digit primes (on a calculator) that remain prime if viewed upside down, in a mirror, or upside down in a mirror. *Prime Curios<br /><br /> There are 52 letters in the names of the cards in a standard deck: ACE KING QUEEN JACK TEN (This also works in Spanish. any other languages for which this is true?) *<a href="http://www.futilitycloset.com/" target="_blank">Futility Closet </a> <br /><br />52 is called an "untouchable" number, since there is no integer for which the sum of its proper divisors sum to 52. Can you find another? Euler said they were infinite.<br /><br />A triangle with sides 51, 52 and 53 has an integer area 1170 units<sup>2</sup>. These are called Heronian Triangles, or sometimes <span style="background-color: white; color: #222222; font-size: 16px;">I call them Sang-Heronian triangles after the earliest study I know about them by Edward Sang of Edinburgh, Scotland in 1864</span>. They have consecutive integer sides Each of these triangles can be partitioned into two Heronian right triangles by the altitude to the even side. It seems that in all such triangles, the altitude will divide the even base into two sides whose lengths differ by 4. For this one, the two right triangle bases will be 26-2 and 26+2. To find the height of the triangle, we use the simple A=1/2 b*h , so 1170 = 26*h, and we get h = 45. So the two right triangles have sides of 24, 45, 51 with area of 540 sq units; and 28, 45, 53 with area of 630 sq units. In every pair of right triangles formed by the altitude, one of them is a Primitive Pythagorean Triangle. In this one the PPT is 28, 45, 53.<br /><hr /><b>The 53rd Day of the Year:</b><br />The 53rd day of the year; the month and day are both prime a total of 53 times in every leap year, but not today.<br /><br /> If you reverse the digits of 53 you get its hexadecimal representation; no other two digit number has this quality. You also get the sum of the divisors of 53^3.<br /><br /> The sum of the first 53 primes is 5830, which is divisible by 53. It is the last year day for which n divides the sum of the first n primes. (what were the others?)<br /><br /><b>53</b> is the sum of <b>5</b> consecutive numbers, with an average interval of <b>3</b><br /><b><br /></b>If you raise 2^n starting at one, and searching for a number with two adjacent zeros, you want find one until n = 53.<br /><br />53 is the smallest prime p such that 1p1 (ie, 1531) , 3p3, 7p7 and 9p9 are all prime.(Can you find the 2nd smallest?) Raj Madhuram suggested 2477 is the second smallest of these, and offered the wonderful term, "Sandwich Primes." (Raj actually found five more four digit primes that are "sandwich-able". We leave them as a challenge for the reader. )<br /><br />53 is the smallest prime number that does not divide the order of any sporadic group *Wik<br /><br />A triangle with sides 51, 52 and 53 has an integer area 1170 units<sup>2</sup>. These are called Heronian Triangles, or sometimes Fleenor-Heronian Triangles, because they have sides of consecutive integers. The even side of these triangles is related to a classic equation from Diophantus' Arithmetic (AD 200's). This one is now known as a type of Pell Equation \( x^2 - 3y^2 = 1\). For example it is easy to see that x=2, y=1 is a solution, and the x=2, doubled becomes the even side in the 3,4,5 Triangle. The triangle with even side of 52, is from the solutions x=26, y=15. If you explore the successive rational convergents to the \(\sqrt{3}\), these occur as every other term in that series. \( \frac{2}{1} , \frac{5}{3}, \frac{7}{4}, \frac{19}{11}, \frac{26}{15}...\).<br /><br />Computer Geeks (the capital shows respect) may know that 53 has a prime ASCII code, 3533. It is the smallest prime for which that is true.<br /><br />The floor function of \(e ^\phi \) is 53.<br /><br />You may know that with the traditional Birthday Problem, 23 people reduces the chance of not finding a match to about 1/2. Increase that number to 53, and the probability of no match is about 1/53. <br /><br />53 is a self number, since it cannot be formed as the sum of any integer and its digits.<br /><br />another from Jim Wilder, the sum of the digits of \( 53^7 = 1174711139837\) is 53.<br /><br /> 53 appears twice in one of the most incredible factorizations I've ever known. The number 13532385396179 has prime factors of 13, 53^2, 3853, and 96179, using exactly the same digits in order when you include the square. *Alon Amit <hr /><b>The 54th Day of the Year: </b><br />54 is the smallest number that can be written as the sum of 3 squares in 3 ways.(<i>Well, go on, find all three ways!</i>)<br />And the 54th Prime Number, is the smallest number expressible as the sum of 3 cubes in 3 ways. *Prime Curios<br /><br />There are 54 ways to draw six circles through all the points on a 6x6 lattice. *gotmath.com<br /><br />54 is the fourth Leyland number, after mathematician Paul Leyland. Leyland numbers are numbers of the form \(x^y + y^x \) where x,y are both integers greater than 1.<br /><br /> And the Sin(54<sup>o</sup>) is one-half the golden ratio. <br /><br />Of course, we should add that the Rubiks Cube has 54 squares.<br /><br /><div class="separator" style="clear: both; text-align: center;"><b 46="" 4="" 5="" a="" and="" base="" br="" in="" is="" nbsp="" palindrome=""><a href="https://1.bp.blogspot.com/-nYsdmSpcTeg/Xj3KO5YjhzI/AAAAAAAALKQ/Ng2ehFiWPhMokVXkMCCvMV2e4ij8vDBmwCLcBGAsYHQ/s1600/Rubiks_cube.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="220" data-original-width="220" src="https://1.bp.blogspot.com/-nYsdmSpcTeg/Xj3KO5YjhzI/AAAAAAAALKQ/Ng2ehFiWPhMokVXkMCCvMV2e4ij8vDBmwCLcBGAsYHQ/s1600/Rubiks_cube.jpg" /></a></b></div><br /> Not sure how he finds these, but Jim Wilder just keeps coming up with them; \(54^6\) = 24794911296, and the sum of those digits is 54. (also see day 53) <hr /><b><br /></b><b>The 55th Day of the Year: </b><br />55 is the largest triangular number that appears in the Fibonacci Sequence. (<i>Is there a largest square number?</i>)<br /><br /> 55 is also a Kaprekar Number: 55² = 3025 and 30 + 25 = 55 (Thanks to Jim Wilder)<br /><br /> And speaking of 5<sup>2</sup>, Everyone knows that 3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup>, but did you know that 33<sup>2</sup> + 44<sup>2</sup> = 55<sup>2</sup> But after that, there could be no more.... right? I mean, that's just too improbable, so why is he still going on like this? You don't think......Nah.<br /><br /><br /> 55 is the only year day that is both a non-trivial base ten palindrome and also a palindrome in base four. <br /><br />Every number greater than 55 is the sum of distinct primes of the form 4n + 3. *Prime Curios <i>Someone help me out here. If this is true, then since 55=37 + 13 + 5 , should this say greater than or equal to 55? </i><br /><br />55 is a square pyramidal number, the sum of the squares of the first 5 positive integers.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-sZELmb1x3uI/XkLqtofajYI/AAAAAAAALNc/UTOXnc853IYtMLzwsLhpYfkzWIH4fbybgCLcBGAsYHQ/s1600/55%2Border%2Bsquared%2Bsquare.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="359" data-original-width="339" height="320" src="https://1.bp.blogspot.com/-sZELmb1x3uI/XkLqtofajYI/AAAAAAAALNc/UTOXnc853IYtMLzwsLhpYfkzWIH4fbybgCLcBGAsYHQ/s320/55%2Border%2Bsquared%2Bsquare.jpg" width="302" /></a></div><br />The first squared square was published in 1938 by Roland Sprague who found a solution using several copies of various squared rectangles and produced a squared square with 55 squares, and side lengths of 4205<br /