Factorial primes

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The Prime Pages

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.

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 34790!-1 142891 p85 May 2002 Factorial
2 26951!+1 107707 p65 May 2002 Factorial
3 21480!-1 83727 p65 Sep 2001 Factorial
4 6917!-1 23560 g1 Oct 1998 Factorial
5 6380!+1 21507 g1 Oct 1998 Factorial
6 3610!-1 11277 C Oct 1993 Factorial
7 3507!-1 10912 C Oct 1992 Factorial
8 1963!-1 5614 CD Oct 1992 Factorial
9 1477!+1 4042 D Dec 1984 Factorial
10 974!-1 2490 CD Oct 1992 Factorial
11 872!+1 2188 D Dec 1983 Factorial
12 546!-1 1260 D Oct 1992 Factorial
13 469!-1 1051 BC Dec 1981 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 multifactorial primes
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
Křížek, M. and Somer, L., "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