# 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).

*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. CrandallandM. A. Penk, "Primes of the formn!± 1 and 2 · 3 · 5^{...}p± 1,"Math. Comp.,38:158 (1982) 639--643. Corrigendum inMath. Comp.40(1983), 727.MR 83c:10006- Borning72
A. Borning, "Some results fork!± 1 and 2 · 3 · 5^{...}p± 1,"Math. Comp.,26(1972) 567--570.MR 46:7133- Caldwell95
C. Caldwell, "On the primality ofn!± 1 and 2 · 3 · 5^{...}p± 1,"Math. Comp.,64:2 (1995) 889--890.MR 95g:11003- CG2000
C. CaldwellandY. Gallot, "On the primality ofn!± 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 ofk! +1 and 2*3*5*^{...}* p +1,"Math. Comp.,34(1980) 303-304.MR 80j:10010

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