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.

(up) 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

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].

(up) Record Primes of this Type

rankprime digitswhowhencomment
19562633# + 1 4151498 p451 Jun 2025 Primorial
27351117# + 1 3191401 p448 Sep 2024 Primorial
36533299# - 1 2835864 p447 Aug 2024 Primorial
46369619# + 1 2765105 p445 Aug 2024 Primorial
56354977# - 1 2758832 p446 Aug 2024 Primorial
65256037# + 1 2281955 p444 Aug 2024 Primorial
74778027# - 1 2073926 p442 Aug 2024 Primorial
84328927# + 1 1878843 p442 Jul 2024 Primorial
93267113# - 1 1418398 p301 Sep 2021 Primorial
101098133# - 1 476311 p346 Mar 2012 Primorial
11843301# - 1 365851 p302 Dec 2010 Primorial
12392113# + 1 169966 p16 Sep 2001 Primorial
13366439# + 1 158936 p16 Aug 2001 Primorial
14145823# + 1 63142 p21 May 2000 Primorial
1542209# + 1 18241 p8 May 1999 Primorial
1624029# + 1 10387 C Dec 1993 Primorial
1723801# + 1 10273 C Dec 1993 Primorial
1818523# + 1 8002 D Dec 1989 Primorial
1915877# - 1 6845 CD Dec 1992 Primorial
2013649# + 1 5862 D Dec 1987 Primorial

(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
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
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