23 (twenty-three) is the natural number following 22 and preceding 24.
In mathematics
Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 and 29; while also being the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23). Twenty-three is also the fifth factorial prime, the second Woodall prime, and a happy number in decimal. It is an Eisenstein prime with no imaginary part and real part of the form It is also the fifth Sophie Germain prime and the fourth safe prime, and the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of 14, 23 is the first Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to for the integers 95, 119, 143, and 529. The third decimal repunit prime after R2 and R19 is R23, followed by R1031.
23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime.
The sum of the first nine primes up to 23 is a square: and the sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713).
23 has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of positive integers (the other is 239). See Waring's problem.
The twenty-third highly composite number 20,160 is one less than the last number (the 339th super-prime 20,161) that cannot be expressed as the sum of two abundant numbers.
23 is the number of trees on 8 unlabeled nodes. It is also a Wedderburn–Etherington number, which are numbers that can be used to count certain binary trees.
The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae.
23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down.
23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem.
23 is the smallest prime such that the largest consecutive pair of smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of smooth numbers.
According to the birthday paradox, in a group of 23 or more randomly chosen people, the probability is more than 50% that some pair of them will have the same birthday. A related coincidence is that 365 times the natural logarithm of 2, approximately 252.999, is very close to the number of pairs of 23 items and 22nd triangular number, 253.
The first twenty-three odd prime numbers (between 3 and 89 inclusive), are all cluster primes such that every even positive integer can be written as the sum of two prime numbers that do not exceed .
The first Mersenne number of the form that does not yield a prime number when inputting a prime exponent is with
On the other hand, the second composite Mersenne number contains an exponent of twenty-three:
Further in the sequence of Mersenne numbers, the 23rd prime number (83) is an exponent to the 14th composite Mersenne number, which factorizes into two prime numbers, the largest of which is twenty-three digits long when written in base ten:
is also twenty-three digits long in decimal, and there are only three other numbers whose factorials generate numbers that are digits long in base ten: 1, 22, and 24.
In geometry
The Leech lattice Λ24 is a 24-dimensional lattice through which 23 other positive definite even unimodular Niemeier lattices of rank 24 are built, and vice-versa. Λ24 represents the solution to the kissing number in 24 dimensions as the precise lattice structure for the maximum number of spheres that can fill 24-dimensional space without overlapping, equal to 196,560 spheres. These 23 Niemeier lattices are located at deep holes of radii in lattice points around its automorphism group, Conway group . The Leech lattice can be constructed in various ways, which include:
By means of a matrix of the form where is the identity matrix and is a 24 by 24 Hadamard matrix (Z/23Z ∪ ∞) with a = 2 and b = 3, and entries X(∞) = 1 and X(0) = -1 with X(n) the quadratic residue symbol mod 23 for nonzero n.