Factorial primes

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The Prime Pages

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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.

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

rank prime digits who when comment

1 308084! + 1 1557176 p425 Jan 2022 Factorial
2 288465! + 1 1449771 p3 Jan 2022 Factorial
3 208003! - 1 1015843 p394 Jul 2016 Factorial
4 150209! + 1 712355 p3 Oct 2011 Factorial
5 147855! - 1 700177 p362 Sep 2013 Factorial
6 110059! + 1 507082 p312 Jun 2011 Factorial
7 103040! - 1 471794 p301 Dec 2010 Factorial
8 94550! - 1 429390 p290 Oct 2010 Factorial
9 34790! - 1 142891 p85 May 2002 Factorial
10 26951! + 1 107707 p65 May 2002 Factorial
11 21480! - 1 83727 p65 Sep 2001 Factorial
12 6917! - 1 23560 g1 Oct 1998 Factorial
13 6380! + 1 21507 g1 Oct 1998 Factorial
14 3610! - 1 11277 C Oct 1993 Factorial
15 3507! - 1 10912 C Oct 1992 Factorial
16 1963! - 1 5614 CD Oct 1992 Factorial
17 1477! + 1 4042 D Dec 1984 Factorial
18 974! - 1 2490 CD Oct 1992 Factorial
19 872! + 1 2188 D Dec 1983 Factorial
20 546! - 1 1260 D Oct 1992 Factorial

The World of Mathematics: Factorial Prime
The Prime Glossary's: Factorial prime
The Prime Glossary's: Multifactorial prime
The chronology of prime number records' Factorial/Primorial Prime Records by year
The Top 20 primorial primes

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