Faculty prime

A factorial prime number is a prime number that is one greater or smaller than a factorial , i.e. a prime number of the form

${\displaystyle n!\;\pm \;1\qquad \quad (n\in \mathbb {N} )}$.

The first faculty primes

The first 10 faculty primes are

2 (0! + 1 or 1! + 1), 3 (2! + 1), 5 (3! - 1), 7 (3! + 1), 23 (4! - 1), 719 (6! - 1), 5039 (7! - 1), 39916801 (11! + 1), 479001599 (12! - 1), 87178291199 (14! - 1), ... (see episode A088054 in OEIS ).

n! - 1 is a prime number for

n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003,… (sequence A002982 in OEIS , resulting in 27 faculty primes ).

n! + 1 is a prime number for

n = 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ... (sequence A002981 in OEIS resulting in 22 factorial primes).

Up to now (as of December 2016) , no other than the listed 49 faculty prime numbers are known.

swell

• Eric W. Weisstein : Factorial Prime . In: MathWorld (English).