A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even).
The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) 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), ...
n! − 1 is prime 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, 632760. (resulting in 28 factorial primes)
n! + 1 is prime for : n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429. (resulting in 24 unique factorial primes - the prime 2 occurs twice)
No other factorial primes are known .
When both n! + 1 and n! − 1 are composite, there must be at least 2n + 1 consecutive composite numbers around n!, since besides n! ± 1 and n! itself, also, each number of form n! ± k is divisible by k for 2 ≤ k ≤ n. However, the necessary length of this gap is asymptotically smaller than the average composite run for integers of similar size (see prime gap).
See also Primorial prime
External links
The Top Twenty: Factorial primes from the Prime Pages Factorial Prime Search from PrimeGrid
References