Fortunate Prime

Let

$$X_k\equiv 1+p_k\char93 ,$$

where $p_k$ is the $k$th Prime and $p\char93$ is the Primorial, and let $q_k$ be the Next Prime (i.e., the smallest Prime greater than $X_k$),

$$q_k=p_{1+\pi(X_k)}=p_{1+\pi(1+p_k\char93 )},$$

where $\pi(n)$ is the Prime Counting Function. Then R. F. Fortune conjectured that $F_k\equiv q_k-X_k+1$ is Prime for all $k$. The first values of $F_k$ are 3, 5, 7, 13, 23, 17, 19, 23, ... (Sloane's A005235), and all known values of $F_k$ 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).

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. 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.