The Top Twenty--a Prime Page Collection

Primorial

This page : Definition(s) | Records | References | Related Pages | RSS 2.0 Feed
  View this page in:   language help
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

(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
11098133# - 1 476311 p346 Mar 2012 Primorial
2843301# - 1 365851 p302 Dec 2010 Primorial
3392113# + 1 169966 p16 Sep 2001 Primorial
4366439# + 1 158936 p16 Aug 2001 Primorial
5145823# + 1 63142 p21 May 2000 Primorial
642209# + 1 18241 p8 May 1999 Primorial
724029# + 1 10387 C Dec 1993 Primorial
823801# + 1 10273 C Dec 1993 Primorial
918523# + 1 8002 D Dec 1989 Primorial
1015877# - 1 6845 CD Dec 1992 Primorial
1113649# + 1 5862 D Dec 1987 Primorial
1213033# - 1 5610 CD Dec 1992 Primorial
1311549# + 1 4951 D Dec 1986 Primorial
146569# - 1 2811 D Dec 1992 Primorial
154787# + 1 2038 D Dec 1984 Primorial
164583# - 1 1953 D Dec 1992 Primorial
174547# + 1 1939 D Dec 1984 Primorial
184297# - 1 1844 D Dec 1992 Primorial
194093# - 1 1750 CD Oct 1992 Primorial
203229# + 1 1368 D Dec 1984 Primorial

(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
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
Chris K. Caldwell © 1996-2014 (all rights reserved)