# 23 - the Ninth Prime Number

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Twenty-three is the ninth prime number, the smallest odd prime which is not a twin prime. Twenty-three is also the fifth factorial prime, the second Woodall prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n Ð²?' 1. In base 10, it is the second Smarandache-Wellin prime, as it is the concatenation of the base 10 representations of the first two primes (2 and 3) and is itself also prime.

The fifth Sophie Germain prime and the fourth safe prime, 23 is 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 14, 23 is a Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to x - ÐŸ†(x) for the integers 95, 119, 143, 529.

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 also has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of integers (the other is 239). See Waring's problem.

23 is a Wedderburn-Etherington number and the sixth happy number. The codewords in the perfect (non-extended) binary Golay code are of size 23.

23 is the first prime P for which unique factorization of cyclotomic integers based on the P'th root of unity breaks down.

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.

There were 23 problems on David Hilbert's famous list of unsolved mathematical problems, presented to the International Congress of Mathematicians in Paris in 1900.

23! is 23 digits long in base 10. There are only three other numbers that have this property: 1, 22, and 24.

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