primorial prime

Clearly the primorial numbers themselves, n#, are rarely prime (in fact just for n = 2 where 2# = 2). So when defining primorial primes authors considered two different flavors--primorials plus one: p#+1 and primorials minus one: p#-1.  We call primes of both of these forms primorial primes.

• 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, 145823, 366439 and 392113 (169966 digits).
• 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 less than 100000 [CG00].

The study of these numbers may have originated with Euclid's proof that there are infinitely many primes which uses p#.

See Also: FactorialPrime, MultifactorialPrime

Related pages (outside of this work)

• Euclid's proof that there are infinitely many primes
• The top twenty primorial primes
• The top twenty factorial primes
• The top twenty multi-factorial primes
• Deficient factorials by Rene Dohmen

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

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.

Templer80

M. Templer, "On the primality of k! + 1 and 2 * 3 * 5 * ^... * p + 1," Math. Comp., 34 (1980) 303-304.  MR 80j:10010