The Top Twenty--a Prime Page Collection

Factorial primes

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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.This page is about one of those forms.

(up) 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].

(up) Record Primes of this Type

rankprime digitswhowhencomment
1308084! + 1 1557176 p425 Jan 2022 Factorial
2288465! + 1 1449771 p3 Jan 2022 Factorial
3208003! - 1 1015843 p394 Jul 2016 Factorial
4150209! + 1 712355 p3 Oct 2011 Factorial
5147855! - 1 700177 p362 Sep 2013 Factorial
6110059! + 1 507082 p312 Jun 2011 Factorial
7103040! - 1 471794 p301 Dec 2010 Factorial
894550! - 1 429390 p290 Oct 2010 Factorial
934790! - 1 142891 p85 May 2002 Factorial
1026951! + 1 107707 p65 May 2002 Factorial
1121480! - 1 83727 p65 Sep 2001 Factorial
126917! - 1 23560 g1 Oct 1998 Factorial
136380! + 1 21507 g1 Oct 1998 Factorial
143610! - 1 11277 C Oct 1993 Factorial
153507! - 1 10912 C Oct 1992 Factorial
161963! - 1 5614 CD Oct 1992 Factorial
171477! + 1 4042 D Dec 1984 Factorial
18974! - 1 2490 CD Oct 1992 Factorial
19872! + 1 2188 D Dec 1983 Factorial
20546! - 1 1260 D Oct 1992 Factorial

(up) Related Pages

(up) 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
Chris K. Caldwell © 1996-2022 (all rights reserved)