Primorial Prime

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, 304723, ... (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, ... (OEIS A005234). The largest known primorial primes as of Aug. 2024 are summarized in the following table (Caldwell).

		digits	discoverer
		63142	May 2000
		158936	Aug. 2001
		169966	Sep. 2001
		1878843	PrimeGrid (Jul. 27, 2024)

It is not known if there are an infinite number of primes for which is prime or composite (Ribenboim 1989, Guy 1994).

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References

Caldwell, C. "On The Primality of

and

." Math. Comput. 64, 889-890, 1995.Caldwell, C. K. "Prime Pages. The Top Twenty: Primorial." http://primes.utm.edu/top20/page.php?id=5.Caldwell, C. "Prime Pages: Database Search." http://primes.utm.edu/primes/search.php?Description=%5E[[:digit:]]1,%23-1&Style=HTML.Caldwell, C. "Prime Pages: Database Search." http://primes.utm.edu/primes/search.php?Description=%5E[[:digit:]]1,%23%2B1&Style=HTML.Caldwell, C. and Gallot, Y. "On the Primality of

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. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 7-8, 1994.

Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/primorial-.ZMudge, M. "Not Numerology but Numeralogy!" Personal Computer World, 279-280, 1997.Pickover, C. A. The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 272-273, 2002.PrimeGrid. "PrimeGrid Primes: Subproject: (PRS) Primorial Prime Search." http://www.primegrid.com/primes/primes.php?project=PRS.PrimeGrid. "Primorial Prime Search Project: Range Statistics." http://www.primegrid.com/stats_prs.php.PrimeGrid. "PrimeGrid's Primorial Prime Search." Sep. 18, 2021. https://www.primegrid.com/download/prs-3267113.pdf.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 4, 1989.Rivera, C. "Problems & Puzzles: Puzzle 010-Primes Associated to Primorials and Factorials." http://www.primepuzzles.net/puzzles/puzz_010.htm.Ruiz, S. M. "A Result on Prime Numbers." Math. Gaz. 81, 269-270, Jul. 1997.Sloane, N. J. A. Sequences A005234/M0669, A006794/M2474, A014545, and A057704 in "The On-Line Encyclopedia of Integer Sequences."Sun, J. "Coordinated Search for Primorial Primes." http://primorialprime.home.comcast.net/.Templer, M. "On the Primality of