## Factorial prime |

No. of known terms | 49 |
---|---|

Conjectured no. of terms | Infinite |

n!±1 | |

First terms | 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 |

Largest known term | 208003!−1 |

A **factorial prime** is a

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

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

*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, ... (resulting in 27 factorial primes)

*n*! + 1 is prime for (sequence A002981 in the

*n*= 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ... (resulting in 21 factorial primes - the prime 2 is repeated)

No other factorial primes are known as of December 2016^{}.

Absence of primes to both sides of a factorial *n*! implies a run of at least 2*n*+1 consecutive *n*! ± *k* is *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

- see also
- external links

Other Languages

dansk: Fakultetsprimtal

Deutsch: Fakultätsprimzahl

français: Nombre premier factoriel

한국어: 계승 소수

italiano: Primo fattoriale

magyar: Faktoriálisprím

日本語: 階乗素数

português: Número primo fatorial

русский: Факториальное простое число

suomi: Kertoma-alkuluku

svenska: Fakultetsprimtal

தமிழ்: காரணீயப் பகாஎண்

Tiếng Việt: Số nguyên tố giai thừa

文言: 階乘質數

中文: 阶乘素数