# Factorial primes

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

### Definitions and Notes

Factorial primes come in two flavors: factorial plus one: n!+1, and factorial minus one: n!-1. The form n!+1 is prime for n=1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477 and 6380 (21507 digits). (See [Borning72], [Templer80], [BCP82], and [Caldwell95].) The form 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 and 6917 (23560 digits). Both forms have been tested to n=10000 [CG2000].

There is more information of primorial and factorial primes in [Dubner87] and [Dubner89a].

### Record Primes of this Type

rankprime digitswhowhencomment
1422429! + 1 2193027 p425 Feb 2022 Factorial
2308084! + 1 1557176 p425 Jan 2022 Factorial
3288465! + 1 1449771 p3 Jan 2022 Factorial
4208003! - 1 1015843 p394 Jul 2016 Factorial
5150209! + 1 712355 p3 Oct 2011 Factorial
6147855! - 1 700177 p362 Sep 2013 Factorial
7110059! + 1 507082 p312 Jun 2011 Factorial
8103040! - 1 471794 p301 Dec 2010 Factorial
994550! - 1 429390 p290 Oct 2010 Factorial
1034790! - 1 142891 p85 May 2002 Factorial
1126951! + 1 107707 p65 May 2002 Factorial
1221480! - 1 83727 p65 Sep 2001 Factorial
136917! - 1 23560 g1 Oct 1998 Factorial
146380! + 1 21507 g1 Oct 1998 Factorial
153610! - 1 11277 C Oct 1993 Factorial
163507! - 1 10912 C Oct 1992 Factorial
171963! - 1 5614 CD Oct 1992 Factorial
181477! + 1 4042 D Dec 1984 Factorial
19974! - 1 2490 CD Oct 1992 Factorial
20872! + 1 2188 D Dec 1983 Factorial

### References

BCP82
J. P. Buhler, R. E. Crandall and M. A. Penk, "Primes of the form n! ± 1 and 2 · 3 · 5 ... p ± 1," Math. Comp., 38:158 (1982) 639--643.  Corrigendum in Math. Comp. 40 (1983), 727.  MR 83c:10006
Borning72
A. Borning, "Some results for k! ± 1 and 2 · 3 · 5 ... p ± 1," Math. Comp., 26 (1972) 567--570.  MR 46:7133
Caldwell95
C. Caldwell, "On the primality of n! ± 1 and 2 · 3 · 5 ... p ± 1," Math. Comp., 64:2 (1995) 889--890.  MR 95g:11003
CG2000
C. Caldwell and Y. Gallot, "On the primality of n! ± 1 and 2 × 3 × 5 × ... × p ± 1," Math. Comp., 71:237 (2002) 441--448.  MR 2002g:11011 (Abstract available) (Annotation available)
Dubner87
H. Dubner, "Factorial and primorial primes," J. Recreational Math., 19:3 (1987) 197--203.
Krizek2008
M. Křížek and L. Somer, "Euclidean primes have the minimum number of primitive roots," JP J. Algebra Number Theory Appl., 12:1 (2008) 121--127.  MR2494078
Templer80
M. Templer, "On the primality of k! + 1 and 2 * 3 * 5 * ... * p + 1," Math. Comp., 34 (1980) 303-304.  MR 80j:10010