素数阶乘素数

素数阶乘素数是形如的素数，其中是素数阶乘。PrimeGrid 正在进行针对此类素数的协同搜索。

当 , 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) 时是素数。这些对应于，其中 , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, ... (OEIS A006794)。已知的最大素数阶乘素数总结在下表中。

		位数	发现者
		6845	1992年12月
		365851	PrimeGrid (2010年12月20日)
		476311	PrimeGrid (2012年3月5日)
		1418398	J. Winskill, PrimeGrid (2021年9月18日)

(也称为欧几里得数) 当 , 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) 时是素数。这些对应于，其中 , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, ... (OEIS A005234)。截至 2024 年 8 月，已知的最大素数阶乘素数总结在下表中 (Caldwell)。

		位数	发现者
		63142	2000年5月
		158936	2001年8月
		169966	2001年9月
		1878843	PrimeGrid (2024年7月27日)

是否有无限多个素数使得是素数或合数尚不清楚 (Ribenboim 1989, Guy 1994)。

另请参阅

阶乘, 阶乘素数, 整数序列素数, 素数阶乘

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参考文献

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