Fortunate Prime
Let
where
is the 
th Prime and 
is the Primorial, and let 
be the Next Prime (i.e., thesmallest Prime greater than 
),
where
is the Prime Counting Function. Then R. F. Fortune conjectured that 
isPrime for all . The first values of 
are 3, 5, 7, 13, 23, 17, 19, 23, ... (Sloane's A005235), and all knownvalues of are indeed Prime (Guy 1994). The indices of these primes are 2, 3, 4, 6, 9, 7, 8, 9, 12, 18, .... Innumerical 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).See also Andrica's Conjecture, Primorial
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 7, 1994. Sloane, N. J. A. SequencesA046066 andA005235/M2418in ``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.
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