Primorial primes are primes of the form 
, where 
is the primorial of 
. A coordinated search for such primes is being conducted on PrimeGrid.
With the empty product convention, the product of the first 0 primes is 1, so adding 1 gives the additional primorial prime 2, corresponding to the 
term in OEIS A014545.
is prime
for 
,
3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132,
85586, 234725, 334023, 435582, 446895, ... (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, 4778027, 6354977, 6533299,
... (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) |
 |  | 2073926 | PrimeGrid (Jul. 31, 2024) |
 |  | 2758832 | PrimeGrid (Aug. 12, 2024) |
 |  | 2835864 | PrimeGrid (Aug. 18, 2024) |
(also known as a Euclid
number) is prime for positive indices 
, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391,
1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865,
637491, ... (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, 5256037,
6369619, 7351117, 9562633, ... (OEIS A005234).
The largest known primorial primes 
as of Oct. 9, 2026 are summarized in the following
table (Caldwell).
 |  | digits | discoverer |
 |  | 63142 | May 2000 |
 |  | 158936 | Aug. 2001 |
 |  | 169966 | Sep. 2001 |
 |  | 1878843 | PrimeGrid (Jul. 27, 2024) |
 |  | 2281955 | PrimeGrid (Aug. 6, 2024) |
 |  | 2765105 | PrimeGrid (Aug. 12, 2024) |
 |  | 3191401 | PrimeGrid (Sep. 14, 2024) |
 |  | 4151498 | PrimeGrid (Jun. 25, 2025) |
It is not known if there are an infinite number of primes 
for which 
is prime or composite
(Ribenboim 1989, Guy 1994).
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. "On the Primality of 
and 
." Math. Comput. 64,
889-890, 1995.Caldwell, C. "Prime Pages: Database Search."
and 
." Math. Comput. 71,
441-448, 2002.Caldwell, C. K. "Prime Pages. The Top Twenty:
Primorial." 
and 
." Math. Comput. 34,
303-304, 1980.