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
1150209! + 1 712355 p3 Oct 2011 Factorial
2147855! - 1 700177 p362 Sep 2013 Factorial
3110059! + 1 507082 p312 Jun 2011 Factorial
4103040! - 1 471794 p301 Dec 2010 Factorial
594550! - 1 429390 p290 Oct 2010 Factorial
634790! - 1 142891 p85 May 2002 Factorial
726951! + 1 107707 p65 May 2002 Factorial
821480! - 1 83727 p65 Sep 2001 Factorial
96917! - 1 23560 g1 Oct 1998 Factorial
106380! + 1 21507 g1 Oct 1998 Factorial
113610! - 1 11277 C Oct 1993 Factorial
123507! - 1 10912 C Oct 1992 Factorial
131963! - 1 5614 CD Oct 1992 Factorial
141477! + 1 4042 D Dec 1984 Factorial
15974! - 1 2490 CD Oct 1992 Factorial
16872! + 1 2188 D Dec 1983 Factorial
17546! - 1 1260 D Oct 1992 Factorial
18469! - 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 © 1996-2014 (all rights reserved)