Primorial

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

(Note that factorial and multifactorial primes now have their own pages.)

Let p# (p-primorial) be the product of the primes less than or equal to p so

• 3# = 2^.3 = 6,
• 5# = 2^.3^.5 = 30, and
• 13# = 2^.3^.5^.7^.11^.13 = 30030.

Primorial primes come in two flavors: primorial plus one: p#+1, and primorial minus one: p#-1. p#+1 is prime for the primes p=2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029 and 42209 (18241 digits). (See [Borning72], [Templer80], [BCP82], and [Caldwell95].) p#-1 is prime for primes p=3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033 and 15877 (6845 digits). Both forms have been tested for all primes p < 100000 [CG00]. 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 5256037# + 1 2281955 p444 Aug 2024 Primorial 2 4778027# - 1 2073926 p442 Aug 2024 Primorial 3 4328927# + 1 1878843 p442 Jul 2024 Primorial 4 3267113# - 1 1418398 p301 Sep 2021 Primorial 5 1098133# - 1 476311 p346 Mar 2012 Primorial 6 843301# - 1 365851 p302 Dec 2010 Primorial 7 392113# + 1 169966 p16 Sep 2001 Primorial 8 366439# + 1 158936 p16 Aug 2001 Primorial 9 145823# + 1 63142 p21 May 2000 Primorial 10 42209# + 1 18241 p8 May 1999 Primorial 11 24029# + 1 10387 C Dec 1993 Primorial 12 23801# + 1 10273 C Dec 1993 Primorial 13 18523# + 1 8002 D Jan 1990 Primorial 14 15877# - 1 6845 CD Dec 1992 Primorial 15 13649# + 1 5862 D Jan 1988 Primorial 16 13033# - 1 5610 CD Dec 1992 Primorial 17 11549# + 1 4951 D Jan 1987 Primorial 18 6569# - 1 2811 D Dec 1992 Primorial 19 4787# + 1 2038 D Jan 1985 Primorial 20 4583# - 1 1953 D Dec 1992 Primorial

Related Pages

• The Prime Glossary's Primorial prime
• The chronology of prime number records' Factorial/Primorial Prime Records by year
• The Top 20 factorial 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

CD93

C. Caldwell and H. Dubner, "Primorial, factorial and multifactorial primes," Math. Spectrum, 26:1 (1993/4) 1--7.

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.

Dubner89a

H. Dubner, "A new primorial prime," J. Recreational Math., 21:4 (1989) 276.

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