Factorial Prime (original) (raw)

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A factorial prime is a prime number of the form n!+/-1 , where is a factorial.

n!-1 is prime for n=3 , 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://t5k.org/primes/page.php?id=121944)

n!+1 is prime for n=1 , 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.

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

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References

Boncompagni, B. "Factorization of Special Form Numbers." http://factors.redgolpe.com/.Borning, A. "Some Results for k!+1 and 2·3·5·p+1 ." Math. Comput. 26, 567-570, 1972.Buhler, J. P.; Crandall, R. E.; and Penk, M. A. "Primes of the Form M!+1 and 2·3·5...p+1 ." Math. Comput. 38, 639-643, 1982.Caldwell, C. K. "Prime Links++." http://primes.utm.edu/links/theory/special_forms/near_products/factorialCaldwell, C. K. "The Top Twenty: Factorial." https://t5k.org/top20/page.php?id=30.Caldwell, C. K. "On the Primality of N!+/-1 and 2·3·5...p+/-1 ." Math. Comput. 64, 889-890, 1995.Dubner, H. "Factorial and Primorial Primes."J. Rec. Math. 19, 197-203, 1987.Carmody, P. http://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" http://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!!!." http://www.primegrid.com/forum_thread.php?id=1336&nowrap=true#26809. Oct. 5, 2010.PrimeGrid. "PrimeGrid Primes: Subproject: (FRS) Factorial Prime Search." http://www.primegrid.com/primes/primes.php?project=FRS.PrimeGrid PRPNet. "Factorial Prime Search - Primes by User." http://prpnet.primegrid.com:12002/user_primes.html.PrimeGrid PRPNet. "Factorial Prime Search - Server Statistics." http://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 k!+1 and ·3·5...p+1 ." Math. Comput. 34, 303-304, 1980.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.