Factorial Prime

A factorial prime is a prime number of the form , where  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, 26951, 34790, 94550, 103040, 147855, 208003, ... (OEIS A002982), the largest of which are summarized in the following table.

	digits	discoverer
	107,707	Marchal, Carmody, and Kuosa (Caldwell; May 2002)
	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://primes.utm.edu/primes/page.php?id=121944)

 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), the largest of which are summarized in the following table.

	digits	discoverer
	107,707	K. Davis (May 24, 2002)
	507,082	PrimeGrid (Jun. 14, 2011)
	712,354	PrimeGrid (Aug. 39, 2013)

A distributed project searching for factorial primes is being conducted on PrimeGrid.

REFERENCES:

Boncompagni, B. "Factorization of Special Form Numbers." https://factors.redgolpe.com/.

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.

Caldwell, C. K. "Prime Links++." https://primes.utm.edu/links/theory/special_forms/near_products/factorial.

Caldwell, C. K. "The Top Twenty: Factorial." https://primes.utm.edu/top20/page.php?id=30.

Caldwell, C. K. "On the Primality of  and ." Math. Comput. 64, 889-890, 1995.

Dubner, H. "Factorial and Primorial Primes." J. Rec. Math. 19, 197-203, 1987.

Carmody, P. https://fatphil.org/Nuutti/.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 7, 1994.

Kuosa, N. "Factorial Prime Search Progress Pages" https://83.143.57.194:16384/Factorial/.

Pickover, C. A. The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 272-273, 2002.

PrimeGrid. "World Record Factorial Prime!!!." https://www.primegrid.com/forum_thread.php?id=1336&nowrap=true#26809. Oct. 5, 2010.

PrimeGrid. "PrimeGrid Primes: Subproject: (FRS) Factorial Prime Search." https://www.primegrid.com/primes/primes.php?project=FRS.

PrimeGrid PRPNet. "Factorial Prime Search - Primes by User." https://prpnet.primegrid.com:12002/user_primes.html.

PrimeGrid PRPNet. "Factorial Prime Search - Server Statistics." https://prpnet.primegrid.com:12002.

Sloane, N. J. A. Sequences A002981/M0908 and A002982/M2321 in "The On-Line Encyclopedia of Integer Sequences."

Temper, M. "On the Primality of  and ." Math. Comput. 34, 303-304, 1980.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.