A factorial prime is a number that is one less or one more than a factorial and is also a prime number. The first few factorial primes are: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 (sequence A088054 in the OEIS). It is conjectured that only for $n = 3$ are both $n! - 1$ and $n! + 1$ both primes.

Factorial primes have a rôle in an argument that 1 is not a prime number. If $n$ is a positive integer and $p$ is a prime number, $n! + p$ is never a prime for $p < n$ because obviously it will be a multiple of $p$ just as $n!$ is. But $n! + 1$ even though it certainly is a multiple of 1, can be a prime, specifically, a factorial prime. (The same is also true if we subtract instead of add).