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Wolfram Web Resources » 13,634 entries Last updated: Mon Oct 30 2017 Created, developed, and nurtured by Eric Weisstein at Wolfram Research | Calculus and Analysis > Special Functions > Factorials > Number Theory > Prime Numbers > Prime Sums and Products > Number Theory > Prime Numbers > Prime Number Sequences > Algebra > Products > # 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, ... (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, ... (OEIS A006794). The largest known primorial primes as of Nov. 2015 are summarized in the following table (Caldwell).
|  | digits | discoverer
|  | 6845 | Dec. 1992
|  | 365851 | PrimeGrid (Dec. 20, 2010)
|  | 476311 | PrimeGrid (Mar. 5, 2012)
(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). SEE ALSO: Factorial, Factorial Prime, Integer Sequence Primes, Primorial 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-.Z Mudge, 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. "Primorial Prime Search." http://www.primegrid.com/forum_thread.php?id=1129#11690. PrimeGrid. "PrimeGrid Primes: Subproject: (PRS) Primorial Prime Search." http://www.primegrid.com/primes/primes.php?project=PRS. PrimeGrid. http://www.primegrid.com/stats_321_llr.php. PrimeGrid PRPNet. "Primorial Prime Search - Primes by User." http://prpnet.primegrid.com:12008/user_primes.html. PrimeGrid PRPNet. "Primorial Prime Search - Server Statistics." http://prpnet.primegrid.com:12008. 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 and ." Math. Comput. 34, 303-304, 1980. Referenced on Wolfram|Alpha: Primorial Prime CITE THIS AS: Weisstein, Eric W. "Primorial Prime." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PrimorialPrime.html # Wolfram Web Resources
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