# 113

This number is a prime.

The smallest three-digit absolute prime. [Richert]

113^{2} = 12769 and its reversal 96721 = 311^{2}. [Friend]

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.

The smallest prime factor of 12345678910111213 (the concatenation of the first thirteen natural numbers). [Russo]

The number of teeth on each of the Photoelectric Sieve gears was a multiple of one of the primes up to 113.

The atomic number of an element temporarily called ununtrium (symbol Uut). [Kulsha]

The Police (and general emergency) telephone number in Italy.

113 = 1 + (sum of first 13 odd primes - 13)/2. [Russo]

1^0+1^0+3^0=3;1^1+1^1+3^1=5;1^2+1^2+3^2=11;1^3+1^3+3^3=29;1^4+1^4+3^4=83; 113 is the smallest (non-trivial) prime for which the sums of the first 5 powers of the digits are all prime. [Rupinski]

The largest of the fifteen prime numbers p that require a prime number of digits when written in any prime base less than or equal to p. It is 1110001 in base 2, 11012 in base 3, 423 in base 5, 221 in base 7, and requires two digits in bases 11, 13, 17,... 113. [Hartley]

The three circularly permuted primes, i.e., 113, 131 and 311, can be concatenated in 3! different ways. After inserting a prime digit as a separator between these concatenated primes, the following four 11-digit primes can be obtained: 11371317311, 31151315113, 13153115113 and 11373117131. Note that the first two numbers are palindromic primes. [Necula]

The smallest prime of the form (p^2 + q^2 + r^2)/3, where p, q, r are 3 consecutive primes. [Pfoertner]

113 is the sum of two consecutive squares: 7^{2} + 8^{2}. [Schlesinger]

It is possible to form seven sums of primes with the prime value of 113 using the first 22 primes: 2+19+31+61 = 3+37+73 = 5+41+67 = 7+47+59 = 11+23+79 = 13+29+71 = 17+43+53 = 113. [Gallardo]

(Joke) Theorem: Any number is prime. Proof: by the alphabet code, the phrase "any number" is 113. [Gallardo]

113 is the smallest prime such that the sum of the squares of the digits (11) is prime and the product of the digits (3) is also prime. Note that the concatenation of the two prime values (11 and 3) gives the prime number 113. [Capelle]

The only solution (in positive integers) to a^2 + b^3 = c^7 is 15312283^2 + 9262^3 = **113**^7.

Longitude 113 degrees West passes through Dolphin Island in Great Salt Lake, Utah. Think of it as a "prime meridian" for the world's saltiest sailors.

The greatest known prime number p for which there is no prime number between p and p + sqrt(p). [Capelle]

The smallest prime that yields another if the sum of digits of its consecutive digits is sandwiched between each of the corresponding consecutive digits (i.e., 113 becomes 12143). [Silva]

The recently published AINUR (the Atlas of Images of Nuclear Rings) in the journal Monthly Notices of the Royal Astronomical Society covers 113 nuclear rings in 107 [also prime] different galaxies. [Post]

The smallest prime number which is the sum of three fourth powers of primes. It is 2^4 + 2^4 + 3^4. [Hartley]

Let ψ(*n*) be the natural logarithm of the least common
multiple of all the numbers less than or equal to *n*
(so ψ(6) = log(60)). The maximum value of
ψ(*n*)/*n* occurs at 113. [RS62] [Caldwell]

The number of associative binary operations on an 3-set, or the number of labeled semigroups with 3 elements. [Post]

The smallest Sophie Germain emirp. [Poo Sung]

The smallest prime *p* that has the same (prime) digital root as the *p*th prime: 113th prime equals 617. [Beedassy]

This prime is the smallest natural number whose square contains 5 different digits. It ends with the prime 13, the smallest natural number whose square contains 3 different digits. [Capelle]

The smallest obtuse-angled prime. [Pol]

113 is the only multidigit prime of the form n!! + π(n)!!. [Firoozbakht]

The smallest near-repdigit prime that remains prime when the digits are interchanged (331) or replaced by their respective complements (997). [Beedassy]

The smallest prime whose number of digits and sum of digits are twin primes. [Silva]

The number of palindromic primes less than one million. [Green]

The smallest actual prime which is also a "semiprime" in base 10 carryless arithmetic. [Post]

A113 is the license plate number on Andy's mom's vehicle in *Toy Story*. [Mia]

The smallest prime (emirp) whose product of digits and sum of digits are primes (twin primes). [Loungrides]

If A = 1, B = 2, C = 3..., Z = 26 then 'PERMUTABLE' is a permutable prime. [Homewood]

π(*n*) is defined as the number of primes less than or equal
to *n*. The prime number theorem states that π(*n*)*ln(*n*)/*n*
approaches 1 as *n* tends to infinity. The maximum value of
this expression, 1.2550587..., occurs at *n* = 113. [Keith]

113 is the smallest emirp whose sum of digits is prime. [Post]

113 divided into (5*71) ≈ π. Note that 113, 5, and 71, are all prime. [Cirillo]

Cannabidiol (CBD) is only one of at least 113 identified cannabinoids in cannabis plants. [Reynolds]

An FBI investigation determined that 113 emails on Clinton’s servers contained information that was classified at the time.

The star Arcturus actually shines 113 times more brightly than our Sun, if you were able to compare them side by side. [Becker]

113^2 + 131^2 = 173^2 + 1, where 113, 131, 173 are three consecutive Sophie Germain primes. Is this unique? [Gaydos]

The smallest emirp that has an emirp number of composites between it and the next prime. [Honaker]

The prime gap from 113 to 127 is a record prime gap. If you concatenate the last two digits of 113 and 127, then you get 1327, which is the start of a record prime gap from 1327 to 1361. [Jacobs]

The highest named prime-numbered chemical element is now Nihonium, atomic number 113. If you replace each letter in its name with its numeric position in the alphabet you get 1498151492113 (a prime number ending in 113). [Gaydos]

If A = 26, B = 25, C = 24, ... , Z = 1, then 'AN EMIRP' is an emirp. [Homewood]

Let 113 = 100 + 10 + 3, then 113(mod 100), 113(mod 10) and 113(mod 3) are all prime. Is there a larger example like this? [Bergot]

There are a record-breaking twenty-one ways to express 113 as (q*r+s)/2, where q, r, s are distinct odd primes. Found by J. K. Andersen. [Rivera]

113 is the first member of the smallest three primes q, r, s > 100 such that 4*q, 3*r, 2*s are three consecutive natural numbers. These three primes, q=113, r=151, s=227, represent a quarter pound, one-third pound, and a half pound, respectively, in terms of grams, rounded to the nearest integer (a pound = 453.59237 grams). [Yosei]

The LPG-30 computer contained 113 electronic vacuum tubes.

π(x)ln(x)/x has its maximum value at x = 113. [Rosser]

The highest scoring NFL game was in 1966 when the Washington Redskins beat the New York Giants 72-41 for a combined score of 113. [Homewood]

Motel room number in the "Alone Again" video by rock group *Dokken*. [Allison]

Christopher McCandless died in an abandoned bus in the Alaska wilderness after living mostly on squirrels, birds, roots and seeds for 113 days. [Krakauer]

There are 113 primes congruent to 3 mod 4 which have no shorter prime congruent to 3 mod 4 subsequence. [Kehowski]

The only known prime p such that there exist 2<=b<=p-1 such that b^(p-1) is congruent to 1(mod p^3). [Xayah]

The asteroid known as 2021 PH27 completes one lap around our star every 113 Earth days. [Bonney]

In 1622, the English oculist Richard Banister wrote that the different diseases of the eyes were 113 in number. [Olry]

A US gallon of mercury weighs over 113 pounds. [Stipe]