A factorial prime is a prime number of the form 
,
where 
is a positive integer and 
is a factorial.
is prime
for 
,
4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610,
6917, 21480, 34790, 94550, 103040, 147855, 208003, ... (OEIS A002982),
with some large known examples summarized in the following table.
 | digits | discoverer |
 | 142,891 | Marchal, Carmody, and Kuosa (Caldwell; May 2002) |
 | 429,390 | D. Domanov/PrimeGrid (Oct. 4, 2010) |
 | 471,794 | J. Winskill/PrimeGrid (Dec. 14, 2010) |
 | 700,177 | PrimeGrid (Aug. 30, 2013) |
 | 1,015,843 | S. Fukui (Jul. 25, 2016; https://t5k.org/primes/page.php?id=121944) |
 | 3,395,992 | R. Propper (submitted Oct. 20, 2024; https://t5k.org/primes/page.php?id=138667) |
is prime
for 
,
2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951,
110059, 150209, ... (OEIS A002981; Wells 1986,
p. 70), with some large known examples summarized in the following table.
 | digits | discoverer |
 | 107,707 | K. Davis (May 24, 2002) |
 | 507,082 | PrimeGrid (Jun. 14, 2011) |
 | 712,355 | R. Dohmen (Oct. 2011; https://t5k.org/primes/page.php?id=102627) |
 | 1,449,771 | R. Dohmen (submitted Jan. 10, 2022; https://t5k.org/primes/page.php?id=133139) |
 | 1,557,176 | R. Propper (submitted Jan. 20, 2022; https://t5k.org/primes/page.php?id=133196) |
 | 2,193,027 | R. Propper (submitted Feb. 21, 2022; https://t5k.org/primes/page.php?id=133604) |
As of Oct. 24, 2026, the largest known factorial prime is 
, and the largest known example of the form 
is 
(Caldwell). The submission dates in the later table
rows are the dates recorded by PrimePages.
A distributed project searching for factorial primes is being conducted on PrimeGrid.
and 
." Math. Comput. 26,
567-570, 1972.Buhler, J. P.; Crandall, R. E.; and Penk, M. A.
"Primes of the Form 
and 
." Math. Comput. 38,
639-643, 1982.Caldwell, C. K. "On the Primality of 
and 
." Math. Comput. 64,
889-890, 1995.Caldwell, C. K. "The Prime Glossary: Factorial
Prime." 
and 
." Math. Comput. 34,
303-304, 1980.