Factorial prime

A factorial prime is a prime number that is one less or one more than a factorial (all factorials > 1 are even). The first few factorial 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), ... (sequence A088054 in OEIS)

n! − 1 is prime for (sequence A002982 in OEIS):

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, ...

n! + 1 is prime for (sequence A002981 in OEIS):

n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ...

No other factorial primes are known as of May 2014.

Absence of primes to both sides of a factorial n! implies a run of at least 2n+1 consecutive composite numbers, since n! ± k is divisible by k for 2 ≤ k ≤ n. However, the necessary length of this run is asymptotically smaller than the average composite run for integers of similar size (see prime gap).
See also

Primorial prime

External links

Weisstein, Eric W., "Factorial Prime", MathWorld.
The Top Twenty: Factorial primes from the Prime Pages
Factorial Prime Search from PrimeGrid

Mathematics Encyclopedia