# Primorial

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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. Comments and suggestions requested.

### 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

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

### 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