For the th Prime ,
is Prime for Primes , 5, 11, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, ... (Sloane's A014563; Guy 1994), or for , 3, 5, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, .... is known to be Prime for the Primes , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, ... (Sloane's A005234; Guy 1994, Mudge 1997), or for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, ... (Sloane's A014545). Both forms have been tested to (Caldwell 1995). It is not known if there are an infinite number of Primes for which is Prime or Composite (Ribenboim 1989).
See also Factorial, Fortunate Prime, Prime Sum Smarandache Near-to-Primorial Function, Twin Peaks
References
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.
Caldwell, C. ``On The Primality of and
.'' Math. Comput. 64, 889-890, 1995.
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 and
ftp://sable.ox.ac.uk/pub/math/factors/primorial+.Z.
Mudge, M. ``Not Numerology but Numeralogy!'' Personal Computer World, 279-280, 1997.
Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 4, 1989.
Sloane, N. J. A. Sequences A002110/M1691, A005234/M0669, A014545, and A014563 in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html.
Temper, M. ``On the Primality of and
.'' Math. Comput. 34, 303-304, 1980.
© 1996-9 Eric W. Weisstein