Primorial primes are primes of the form , where is the primorial of . A coordinated search for such primes is being conducted on PrimeGrid.
is prime for , 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, 234725, ... (OEIS A057704; Guy 1994, pp. 7-8; Caldwell 1995). These correspond to with , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, ... (OEIS A006794). The largest known primorial primes are summarized in the following table.
digits | discoverer | ||
6845 | Dec. 1992 | ||
365851 | PrimeGrid (Dec. 20, 2010) | ||
476311 | PrimeGrid (Mar. 5, 2012) | ||
1418398 | J. Winskill, PrimeGrid (Sep. 18, 2021) |
(also known as a Euclid number) is prime for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, ... (OEIS A014545; Guy 1994, Caldwell 1995, Mudge 1997). These correspond to with , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, ... (OEIS A005234). The largest known primorial primes as of Aug. 2024 are summarized in the following table (Caldwell).
digits | discoverer | ||
63142 | May 2000 | ||
158936 | Aug. 2001 | ||
169966 | Sep. 2001 | ||
1878843 | PrimeGrid (Jul. 27, 2024) |
It is not known if there are an infinite number of primes for which is prime or composite (Ribenboim 1989, Guy 1994).