Factorial prime

Last updated
Factorial prime
No. of known terms53
Conjectured no. of termsInfinite
Subsequence ofn! ±1
First terms2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199
Largest known term632760!  1
OEIS index A088054

A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). [1]

Contents

The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (sequence A088054 in the OEIS ):

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 (sequence A002982 in the 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, 208003, 632760. [2] (resulting in 28 factorial primes)

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

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 as of June 2025.

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

References

  1. "Weisstein, Eric W. "Factorial Prime." From MathWorld".
  2. Caldwell, Chris. "Factorial primes". The Prime Pages. Retrieved 2025-06-29.