Saturday, March 21, 2020

Number Facts for Every Year Day (91-120)

The 91st Day of the Year:
The 91st day of the year; 10n + 91 and 10n + 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?)

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.

91 is : The sum of thirteen consecutive integers = 1 + 2 + 3 + ... + 11 + 12 + 13, and thus the thirteenth triangular number.
and of six consecutive squares= 12 + 22 + 32 + 42 + 52 + 62  making it a pyramidal number,

91 is the sum of two consecutive cubes = 33 + 43 and the difference of two consecutive cubes = 63 - 53

91 is also the sum of three squares, 1^2 + 3^2 + 9^2 .

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)

91 = 46^2 - 45^2 = 10^2 - 3^2

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. 

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

91 is a repdigit in base 9 (111) (9^2+9+1) and a palindrome in base 9 and base 3 (10101) (3^4 + 3^2 + 1)

91 is the smallest pseudoprime (two unique prime factors) for which it is true that 3<sup>n</sup> = 3 mod n *Prime Curios

Prime numbers less than 10,000,000 occur with the two digit ending 91 more than any other ending.  *Prime Curios

The 92nd Day of the Year:
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.  *Prime Curios  for instance in base 8 it is expressed as \(431_8 \) and it's reversal,  \(134_8 \) =89

 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.
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)

 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)

92 is a palindrome in bases 6 (2326), and 7 (1617)

Unlike 91 (and lots of other numbers) 92 can not be written as the sum of three positive squares.

Because 92 is divisible by four, it is the difference of two squares, 24^2 - 22^2

The 93rd Day of the Year:
The first 93 digits of 93! form a prime number. *Prime Curios 
93! = 1156772507081641574759205162306240436214753229576413535186142281213246807121467
315215203289516844845303838996289 ...

93 is a Plum Integer since both its divisors, 3 and 31, are Gaussian Primes (Primes of the form 4n+3)

93 is the sum of three distinct squares, 93 = 22 + 52+ 82 and six consecutive integers 93= 13 + 14 + 15 + 16 + 17 + 18  (another trapezoidal number, see (92nd Day)

There are 93 five-digit prime palindromes. The smallest (I think) is 10301

A potato can be cut into 93 pieces with just nine straight cuts.

and 93 in base 10 is 333 in base 5

93, 94, and 95 form the third string of three consecutive semiprimes (two distinct factors)

93 is a palindrome in base 2, \( 1011101_2\) and in base 5 \( 333_5\)

There are 93 different real periodic points of order 11 on the Mandelbrot set.
260 - 93 is prime. The statement is untrue if 93 is replace with any integer smaller than 93

93 = 47^2 - 46^2 = 17^2 - 14^2

The 94th Day of the Year
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.

94 begins the smallest string of three consecutive numbers none of which is a palindrome in any base, b \( 2 \leq b \leq 10 \)

Add the prime factors of 94 and the result is 49, 94 reversed.

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)

94 is the smallest even number greater than four which cannot be written as a sum of two twin primes.  *Prime Curios

94 has all square digits,  The 94th prime is 491, also with all square digits.

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.

and 94 is the 29th semiprime, and the fourteenth with 2 as one of those two prime factors.

1100977 and 1101071 are a pair of consecutive primes. They form the last pair of primes known that are two digits (94) apart.

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.  942= \$ 8836_{10} = 673_{11}\$ and \$ 6^2 + 7^2 + 3^2 = 94 \$ *Wik

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   \(64^3 + 94^3 = 103^3 + 1^3  \)


The 95th Day of the Year:
950 + 951 + 952 + 953 + 954 + 955 + 956 is prime. *Prime Curios

95 and its reversal (59) begin fewer four-digit prime numbers (seven) than any other two-digit number.

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
4 2 2 2
3 2 2 2 1

 95 is the sum of 7 consecutive primes = 5 + 7 + 11 + 13 + 17 + 19 + 23

NINETYFIVE is the largest semiprime that can be spelled with a semiprime number of toothpicks. *Prime Curios

95 is the third member of the third sequence of three consecutive semiprimes.

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.)

The 96th Day of the Year:

96 is the smallest number that can be written as the difference of 2 squares in 4 ways. *What's So Special About This Number?   (students are encouraged to find them all...Is there a smaller number that can be so expressed in 3 ways?)

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

96 is a strobogrammatic number, rotated 180 degrees it is still 96.

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

Ninety Six, South Carolina. Ninety Six figured prominently in the Anglo-Cherokee War (1758–1761). During the American Revolutionary War, it was a site for southern campaigns. The first land battle of the revolution south of 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.

\$ \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.

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.

The 97th Day of the Year:
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?) *Prime Curios

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. 

 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 92 + 72  > 97

There are 97 leap days every 400 years in the Gregorian Calendar

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

The smallest prime that has a prime alphabetic value in its Roman numerals based representation, i.e., XCVII -> 24 + 3 + 22 + 9 + 9 = 67.

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

Now add
 look at 1/997.

If  your watching your weight, a jigger (1.5 oz) of most spirits contain about 97 calories. Please blame *Prime Curios

between 100 and 1000, there are 97 primes with distinct digits.

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

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.

\$97 = 2^4 + 3^4 \$  the sum of two consecutive primes to the same power.  It is the largest known prime with this property.. 

 97, 907, 9007, 90007 and 900007 are all primes.

The 98th Day of the Year:
98 is the smallest number that starts a sequence of three consecutive numbers with at least 3 prime divisors. (What would be the smallest number to start a sequence of four numbers with at least four prime divisors?)

98 is the sum of fourth powers of the first three integers, 14 + 24 + 34  Only one larger year day is the sum of the first 3 nth powers .

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?)

98 is a ambinumeral, rotating it 180 degrees produces another integer, 86.

98 is a palindrome in base 5 (343) , and base 6 (242)

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.  

The 99th Day of the Year:
If 99 divides some 4-digit number ABCD, then 99 also divides BCDA, CDAB, and DABC

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 is related to a conjecture of Lemoine. 
The conjecture states that any odd number greater than 5 can be written as p+2q 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.)  

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

99 is the sum of the cubes of three consecutive numbers, \( 2^3 + 3^3 + 4^3 \)

99 is the sum of all the sums of all the divisors (including themselves) from one to 11.*Wik

and 99 is the alphanumeric value of THIRTEEN *Number Gossip

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.

992 = 9801 and 98 + 01 = 99 so it is a Kaprekar number, named after D. R. Kaprekar, an Indian recreational mathematician.

and David Marain ‏@dmarain recently reminded me 1/992 = It is the eleventh Year Day that has this quality, but there are only 9 more for the rest of the year.  

0.000102030405060708091011121314151617181920212223242526272... The question for students, It must be a repeating decimal, when does it start to repeat?

(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...)

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.  

The 100th Day of the Year: 
The first 3 primes add to 10 and the first 32 primes add to 102 = 100 *Prime Curios
and the cube of the first four positive integers sum to 100, 1³ + 2³ + 3³ + 4³  *Jim Wilder

And Hansrudi Widmer tweeted that 100 = 2⁶ + 6². 

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) .
Hey! Can you make 100 in my Five Factorials Game.  Use 1!, 2!, 3!, 4!, 5! and only the operations +, -, *, and /

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."

100 years is 3.1536 × 10^9.  so 1 nanocentury is abut pi seconds

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)

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.

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. 

The 101st Day of the Year:
101 is a self-strobogramatic number. It is the second smallest Prime self-strobogrammatic prime, after 11

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

3! - 2! + 1! = 5 (prime) 4! - 3! + 2! - 1! = 19 (prime) 5! - 4! + 3! - 2! + 1! = 101 (prime) HT to Ed Southal   (What would be the next number created in a sequence like this? Is it prime?)

101 is the sum of five consecutive primes, It is the fourth prime year day that is the sum of five consecutive primes.

 101 = 5! - 4! + 3! - 2! + 1!

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."

101 is the largest known prime of the form 10n + 1.

 There are 101 digits in the product of the 39 successive primes produced by the formula n2 + 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

 and The last five digits of 101101 are 10101.And.. the 101 Fibonacci number ends in 101, also.

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
Folks in Kentucky know that  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'.

Take any four digit repdigit, like 7777, and hey, its largest prime divisor is 101.     *Prime Curios

The sum of the squares of all the prime numbers up to 101 is prime.... 

and 101 is a palindrome in base ten, but not in any smaller base.  

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.

Research into possible odd perfect numbers has revealed that the largest factors must be greater than  101, 10007, and 10000007.  *Wolfram MathWorld  

The 102nd Day of the Year 
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.
 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.)
 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)

Another writer wrote to tell me that 102 is the sum of four consecutive primes, 102 = 19 + 23 + 29 + 31,
And there used to be a US 102 in Michigan, but they did away with the name, and the road is now US 142.

The 103rd Day of the Year
there are 103 geometrical forms of magic knight's tour of the chessboard. 

103 is the reverse of 301. The same is true of their squares: 1032 = 10609 and 3012 = 90601. *Jim Wilder

 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. )

 Using a standard dartboard, 103 is the smallest possible prime that cannot be scored with two darts.

If you concatenate the numbers from 103 down to one (10310210101100....) the result is divisible by 101
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

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,

In yesterdays notes I pointed out that US 102 no longer exists. As far as I can find, US 103 never did.  ANYONE?

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   \(64^3 + 94^3 = 103^3 + 1^3  \)

The 104th Day of the Year
104 is the smallest known number of unit line segments that can exist in the plane, 4 touching at every vertex. *What's Special About This Number

104 is the sum of eight consecutive even numbers, 104 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20

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

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,... the differences give a clue to the complete pattern.)

104 is a palindrome in base five (404), and in base six (252), and base twelve (88)

There are 588939451 left and right truncatable primes (truncate the two outsde digits at once) with an even number of digits. The largest is the 104-digited prime number 91617596742869619884432721391145374777686825634291523771171391111313737919133977331737137933773713713973.

and look at your keyboard, the standard Windows keyboard has 104 keys

104+1^3+0^3+4^3 = 13^2 . (n + sum of cubes of its digits is a square)  There is one smaller year day with this quality, and it's prime.

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.

The 105th Day of the Year
Paul Erdős conjectured that this is the largest number n such that the positive values of n - 2k are all prime. *Prime Curios Can you find a smaller number for which this is true?

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

As the sum of the first fourteen integers, it is a Triangular number.

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.

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.

105 is the largest number for which all the odd composite numbers less than it,either share a factor with it, or are prime.

105 is a palindrome in base four (1221) and in base eight (151) and base twenty (55).

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. 

The 106th Day of the Year
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.

106106-105105 (a number of 215 decimal digits)is prime.

 There are 106 distinct mathematical trees with ten vertices.

106 is the fifteenth day of the year that in the form 2P where P is a prime.

 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.

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.

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.

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. 

The 107th Day of the Year

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.*Prime Curios (This seems a most remarkable set of facts to me.)

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?

2107 - 1 is the largest known Mersenne prime not containing all the individual digits. This number is a 33 digits long; 162259276829213363391578010288127.  .  

 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.

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

107 remains prime when 2 is added to any one of its digits. 307. 127, and 109 are all prime.

107 is the sum of three prime numbers that all end in a 9.  19 + 29 + 59 = 107

107 +n! is prime for all single digits greater than one.

107 and its reversal, 701 are both prime.  Such pairs are generally called emirps

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.

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.  

The 108th Day of the Year
For fans of Pentominoes, you may try to construct all the Heptominoes, (made of 7 squares).  There are 108 of them.

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

AND 108 = 1¹ • 2² • 3³ *jim wilder ‏@wilderlab  Numbers like this are called Hyperfactorials, this one is the third.

The concatenation of 108 with its previous and next number is prime, i.e., 108107 and 108109 are primes.

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.)

108 is also, as every good geometry student knows, the interior angle measures of each angle of a regular pentagon

According to Vedic cosmology, 108 is the basis of creation, represents the universe and all our existence. 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

Today and tomorrow are both examples of ambinumerals,or invertibl numbers, which form a different number when rotated 180o 108 becomes 801. Numerals like 181 which stay the same when rotated are called strobogrammatic numerals. This distinction is not always maintained.

The 109th Day of the Year,

109 is the 29th prime, and the tenth superprime, a prime number whose prime rank, or index,(29)  is also prime.

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?)

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.

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..
5 * 7 = 35, 3 + 5 = 8
11 * 13 = 143, 1 + 4 + 3 = 8
17 * 19 = 323, 3 + 2 + 3 = 8
Hat tip to Ben Vitale

109 = 1*2+3*4+5*6+7*8+9.

109 is the smallest prime which is half the difference of two cubes, (7^3-5^3)/2

The period of the reciprocal of 109 ends with 853211 (the beginning of the Fibonacci sequence reversed).

109 rotated 180o 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.

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

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.

109 is the product of two primes raised to their own power, \$ 109 = 2^2 * 3^3\$

109 is remembered for being the number of the PT boat that future President Kennedy was on when it was hit in WWII.

110th Day of the Year
110 is the average of first fifty-three primes.

 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. *

110 = 5^2 + 6^2 + 7^2 (3 consecutive squares) = 11^2 - 11^1 (difference between powers of the same number)

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?

110 hertz is the standard frequency of the musical note A or La.

110 is also known as "eleventy" according to the number naming system invented by J. R. R. Tolkien.

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.

And 110 is the most commonly used impossible percentage, "Give 110%."

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

111th Day of the year,

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.

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)
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

Numbers like 111 that appear the same under 180o 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".

 If you concatenated three copies of 111 and then squared the result, you get (111,111,111)2 = 12,345,678,987,654,321 *Cliff Pickover@pickover

 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.

111 is the sum of the non-prime numbers from 2 through 17. *Prime Curios

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!.

111 is the smallest palindrome that has a prime digit sum.

There are exactly 111 prime numbers that display on a digital clock.  *Prime Curios

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)  .

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.

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.

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.  

The 112th Day of the Year:
112 is a practical number (aka panarithmetic numbers), any smaller number can be formed with distinct divisors of 112.  Student's might explore the patterns of such numbers.

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)

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!(1+1+2)+1 is prime.

112 = 11 + 13 + 17 + 19 + 23 + 29 (sum of consecutive primes) and = 1x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 (sum of oblong or pronic numbers)

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

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

There are 112 pounds in a British long hundredweight.

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.
Of all the Harshad numbers, few of them are also divisible by the product of their digits, and all of the year days that meet both conditions are divisible by three.  Only two year days are not.  One is 112, enjoy finding the other.

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

The 113th Day of the Year;
113 is prime, its reversal (311) is prime, and the number you get by any reordering of its digits is still prime. Students might try to find other of these "absolute" or "permutable" primes.  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.

 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???)

 \$ 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, "Zu Chongzhi (or Tsu Ch'ung Chi), along with his son Zu Gengzhi, stated in a mathematical text titled Zhui Shu (Method of Interpolation) that π is approximately three hundred fifty-five divided by 113." *Prime Curios
 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 to pi for six digits, 3.1415929.... The error is less than (1/113)^2.  

In the infinity of integers, there is only one solution to n^7 = p^2 + q^3 : 113^7= 15312283^2 + 9262^3

There are 13 consecutive divisible integers (non-primes) between 113 and 127. How far until the next streak as long, or longer?

113 was once the Atomic name of Element 113, ununtrium, which was later renamed Nihonium for the country of its discoverers, although a Russian team was also considered for the discovery.  

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

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.  

113 is the largest known prime, P, for which there is no prime between P and P+sqrt(P) = (113-123)  

113 is a palindrome in base eight, 161.  

113 is the smallest of five consecutive primes whose sum is prime.  

113 is the smallest integer that can not be represented in the Four-fours game under the standard rules.  

113 is the largest number in a 3x3 prime magic square that has the smallest possible total for each row, column and diagonal, 

The 114th Day of the Year
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

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.

The largest gap between two consecutive six digit primes is 114.

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.

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.

114 is a repdigit in base 7 (222) and a palindrome in base 5 (424)

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. 

The 115th Day of the Year
115 is the 26th "Lucky" number. Lucky numbers 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.

 \( \pi (115) = 30 \) 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.
Another way to get 30, is  /(  (1*1*5)^1 + (1*1*5)^2  /)

 There are 115 ways (without including rotations and reflections) of placing 6 rooks on  a standard chessboard so that they are not attacking

The 116th Day of the Year
116! +1 is prime.  It is the 10th such year day, but only the second even number with the attribute.
 And: 116^2 + 1 is prime

The number 1 appears 116 times in the first 1000 digits of pi. Thanks to *Math Year-Round ‏@MathYearRound

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.  

Impress your History teacher, the 100 Years war between France and England..... lasted 116 years. 

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.

 And for a bit of Americana, from a British web site called *  It's about Hyperion, a tree that is 116 meters tall.

The 117th Day of the Year

117 is the smallest possible length of the longest side of a Heronian tetrahedron (one whose sides, face areas, and Volume are all integers) *Prime Curios
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.

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.  

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. 

117 can be written as the difference of two squares, or of two cubes.  \$11^2 - 2^2 = 5^3 - 2^3 = 117 \$  (Can you find another number which can be expressed as both the difference of squared primes and cubed primes?)

117 is a repdigit in base 12(99) and a palindrome in base 6(313).

117 is the sum of three consecutive powers of three, 3^2 + 3^3 + 3^4 = 117

The 118th Math Day of the Year
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.

118 is the smallest even number not differing by one or a prime number from one of its prime neighbors.
*Prime Curios

There are four unique partitions of 118 into three integers that all have the same product.  No smaller example exists.  14 × 50 × 54 = 15 × 40 × 63 = 18 × 30 × 70 = 21 × 25 × 72 = 37800. 

In the spelling, ONE HUNDRED EIGHTEEN, one hundred uses 11 character spaces, Eighteen uses 8 , concatenated, we have 118.

The 119th Math Day of the Year
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)

119 is the product of the first two primes ending with 7

119 is a palindrome in base 2, (1110111), and in base 16, (77)

119 is the order of the largest cyclic subgroups of the Monster group.

There are 119 prime numbers which gets displayed on a 12-hour digital clock

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?

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)

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,

The 120th Math Day of the Year
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

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)

120 = 3¹ + 3² + 3³ + 3⁴

Had to add this one,120 is the smallest number to appear 6 times in Pascal's triangle. *What's Special About This Number
(There are only three days of the year that appear in the arithmetic triangle more than five times. What are the other two?)

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)

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.

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. 

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.

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.  

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)!.

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:

1.........32........243.........1024........... 3125.........7776........16807
............180.........570.........1320.........2550......   4380
....................390 ........750............1230.......1830
And in general, the nth differences of the nth powers of the integers is n!  *Conway and Guy, The Book of Numbers

120 is the sum of a twin prime pair, (59+61) 

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 =  1 x 2 x 3 x 4 x 5.  

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 camera when it's 75 years old.  

The smallest number that is the product of reversible number  in two different ways is 2520 = 120 x 021 = 210 x 012.  
The smallest with no zeros is   144648 = 861 x 168 = 492 x 294.  The two smaller numbers in each pair (168 &264) are called Reversible product twins. 
The number 144646 is called a double reversible number.  Sometimes one of the two numbers is a palindrome, such as, 63504 = 252^2 = 441 x 144.  This one is interesting because it involves all so many square numbers. 

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