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. Comments and suggestions requested.

(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
1208003! - 1 1015843 p394 Jul 2016 Factorial
2150209! + 1 712355 p3 Oct 2011 Factorial
3147855! - 1 700177 p362 Sep 2013 Factorial
4110059! + 1 507082 p312 Jun 2011 Factorial
5103040! - 1 471794 p301 Dec 2010 Factorial
694550! - 1 429390 p290 Oct 2010 Factorial
734790! - 1 142891 p85 May 2002 Factorial
826951! + 1 107707 p65 May 2002 Factorial
921480! - 1 83727 p65 Sep 2001 Factorial
106917! - 1 23560 g1 Oct 1998 Factorial
116380! + 1 21507 g1 Oct 1998 Factorial
123610! - 1 11277 C Oct 1993 Factorial
133507! - 1 10912 C Oct 1992 Factorial
141963! - 1 5614 CD Oct 1992 Factorial
151477! + 1 4042 D Dec 1984 Factorial
16974! - 1 2490 CD Oct 1992 Factorial
17872! + 1 2188 D Dec 1983 Factorial
18546! - 1 1260 D Oct 1992 Factorial
19469! - 1 1051 BC Dec 1981 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 © 1996-2016 (all rights reserved)