23

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Summary

Factorization

The number 23 is a prime number.

Properties and families

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	ninth prime number	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, [SHOW MORE] 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 View list on OEIS	A natural number $n > 1$ is prime if and only if $n$ is not divisible by any prime less than or equal to $\sqrt{n}$ . In this case, since $\sqrt{23}$ is between 4 and 5, verifying primality requires verifying that 23 is not divisible by any prime up to 4, i.e., it is not divisible by 2 or 3.
safe prime (prime $p$ such that $(p - 1) / 2$ is prime)	fourth safe prime	5, 7, 11, 23, 47, 59, 83, 107, 167, [SHOW MORE] 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903 View list on OEIS	$(23 - 1) / 2$ equals 11, which is prime
Sophie Germain prime (prime $p$ such that $2 p + 1$ is prime)	fifth Sophie Germain prime	2, 3, 5, 11, 23, 29, 41, 53, 83, 89, [SHOW MORE] 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559 View list on OEIS	$2 (23) + 1$ equals 47, which is prime
factorial prime (prime that is of the form factorial $\pm 1$ )	fifth factorial prime	2, 3, 5, 7, 23, 719, 5039, [SHOW MORE] 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 View list on OEIS	$23 = 24 - 1$ and 24 equals $4!$

Polynomials

Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 23 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 23
$n^{2} - n + 11$	2	all numbers 1-10, because 11 is one of the lucky numbers of Euler.	4
$n^{2} - n + 17$	2	all numbers 1-16, because 17 is one of the lucky numbers of Euler.	3

Irreducible polynomials by Cohn's irreducibility criterion

By Cohn's irreducibility criterion, we know that if we write 23 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here the irreducible polynomials of this type that are of degree at least two:

Base $b$	17 in base $b$	Corresponding irreducible polynomial
2	10111	$x^{4} + x^{2} + x + 1$
3	212	$2 x^{2} + x + 2$
4	113	$x^{2} + x + 3$