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, ... (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, ... (OEIS A005234).
The largest known primorial primes 
as of Nov. 2015 are summarized in the following
table (Caldwell).
 |  | digits | discoverer |
 |  | 63142 | May 2000 |
 |  | 158936 | Aug. 2001 |
 |  | 169966 | Sep. 2001 |
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. 64,
889-890, 1995.Caldwell, C. K. "Prime Pages. The Top Twenty:
Primorial." 
and 
." Math. Comput. 71,
441-448, 2002.Borning, A. "Some Results for 
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.Dubner, H. "Factorial and Primorial Primes."
J. Rec. Math. 19, 197-203, 1987.Dubner, H. "A New
Primorial Prime." J. Rec. Math. 21, 276, 1989.Guy,
R. K. 
and 
." Math. Comput. 34,
303-304, 1980.