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 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (sequence in the OEIS):

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

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

No other factorial primes are known as of December 2016^[update].

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).

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