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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) 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
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