Let

where is the th Prime and is the Primorial, and let be the Next Prime (i.e., the smallest Prime greater than ),

where is the Prime Counting Function. Then R. F. Fortune conjectured that is Prime for all . The first values of are 3, 5, 7, 13, 23, 17, 19, 23, ... (Sloane's A005235), and all known values of are indeed Prime (Guy 1994). The indices of these primes are 2, 3, 4, 6, 9, 7, 8, 9, 12, 18, .... In numerical order with duplicates removed, the Fortunate primes are 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, ... (Sloane's A046066).

**References**

Guy, R. K. *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, p. 7, 1994.

Sloane, N. J. A. Sequences
A046066 and
A005235/M2418
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.

© 1996-9

1999-05-26