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Primorial Prime


Primorial primes are primes of the form p_n#+/-1, where p_n# is the primorial of p_n. A coordinated search for such primes is being conducted on PrimeGrid.

p_n#-1 is prime for n=2, 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 p#-1 with p=3, 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 p_n#-1 are summarized in the following table.

p_n#-1p#-1digitsdiscoverer
p_(1849)#-115877#-16845Dec. 1992
p_(67132)#-1843301#-1365851PrimeGrid (Dec. 20, 2010)
p_(85586)#-11098133#-1476311PrimeGrid (Mar. 5, 2012)
p_(234725)#-13267113#-11418398J. Winskill, PrimeGrid (Sep. 18, 2021)

p_n#+1 (also known as a Euclid number) is prime for n=1, 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 p#+1 with p=2, 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 p_n#+1 as of Aug. 2024 are summarized in the following table (Caldwell).

p_n#+1p#+1digitsdiscoverer
p_(13494)#+1145823#+163142May 2000
p_(31260)#+1366439#+1158936Aug. 2001
p_(33237)#+1392113#+1169966Sep. 2001
p_(304723)#+14328927#+11878843PrimeGrid (Jul. 27, 2024)

It is not known if there are an infinite number of primes p for which p#+1 is prime or composite (Ribenboim 1989, Guy 1994).


See also

Factorial, Factorial Prime, Integer Sequence Primes, Primorial

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References

Caldwell, C. "On The Primality of n!+/-1 and 2·3·5...p+/-1." 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 n!+/-1 and 2×3×5×...×p+/-1." Math. Comput. 71, 441-448, 2002.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.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.Update a linkLeyland, 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 k!+1 and 2·3·5...p+1." Math. Comput. 34, 303-304, 1980.

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Primorial Prime

Cite this as:

Weisstein, Eric W. "Primorial Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimorialPrime.html

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