The smallest three-digit absolute prime. [Richert]
1132 = 12769 and its reversal 96721 = 3112. [Friend]
The smallest three digit prime whose product and sum of digits
is prime. [Russo]
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  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: 72 + 82. [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+49+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 smallest prime whose number of digits and sum of digits
are twin primes. [Silva]
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 pth 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 emirp with both the sum of digits and the number
of digits as primes. [Post]
The smallest obtuse-angled
prime. [Pol]
113 is the only multidigit prime of the form n!! + (n)!!. [Firoozbakht]
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