The th Euclid number is defined by

where is the th Prime. The first few are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (Sloane's A006862). The largest factor of are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (Sloane's A002585). The of the first few Prime Euclid numbers are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, ... (Sloane's A014545) up to a search limit of 700. It is not known if there are an Infinite number of Prime Euclid numbers (Guy 1994, Ribenboim 1996).

**References**

Guy, R. K. *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, 1994.

Ribenboim, P. *The New Book of Prime Number Records.* New York: Springer-Verlag, 1996.

Sloane, N. J. A. Sequences
A014545,
A006862/M2698, and
A002585/M2697,
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Wagon, S. *Mathematica in Action.* New York: W. H. Freeman, pp. 35-37, 1991.

© 1996-9

1999-05-25