Primorial

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The Prime Pages

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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

rank prime digits who when comment

1 3267113# - 1 1418398 p301 Sep 2021 Primorial
2 1098133# - 1 476311 p346 Mar 2012 Primorial
3 843301# - 1 365851 p302 Dec 2010 Primorial
4 392113# + 1 169966 p16 Sep 2001 Primorial
5 366439# + 1 158936 p16 Aug 2001 Primorial
6 145823# + 1 63142 p21 May 2000 Primorial
7 42209# + 1 18241 p8 May 1999 Primorial
8 24029# + 1 10387 C Dec 1993 Primorial
9 23801# + 1 10273 C Dec 1993 Primorial
10 18523# + 1 8002 D Dec 1989 Primorial
11 15877# - 1 6845 CD Dec 1992 Primorial
12 13649# + 1 5862 D Dec 1987 Primorial
13 13033# - 1 5610 CD Dec 1992 Primorial
14 11549# + 1 4951 D Dec 1986 Primorial
15 6569# - 1 2811 D Dec 1992 Primorial
16 4787# + 1 2038 D Dec 1984 Primorial
17 4583# - 1 1953 D Dec 1992 Primorial
18 4547# + 1 1939 D Dec 1984 Primorial
19 4297# - 1 1844 D Dec 1992 Primorial
20 4093# - 1 1750 CD Oct 1992 Primorial

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