primorial prime (another Prime Pages' Glossary entries)
 Glossary: Prime Pages: Top 5000: 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, MultifactorialPrimeRelated 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 DohmenReferences: 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 Chris K. Caldwell © 1999-2018 (all rights reserved)