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Fortunate Prime
Consider the Euclid numbers defined by
 |
where 
is the 
th
prime and 
is the primorial. The first
few values of 
are 3, 7, 31, 211, 2311, 30031, 510511, ... (OEIS A006862).
Now let 
be the next prime (i.e., the smallest prime
greater than 
),
 |
where 
is the prime counting function. The first
few values of 
are 5, 11, 37, 223, 2333, 30047, 510529, ... (OEIS A035345).
Then R. F. Fortune conjectured that 
is prime for all
. The first values of 
are 3, 5, 7, 13, 23, 17, 19, 23, ... (OEIS A005235),
and values of 
up to 
are indeed prime (Guy 1994), a result extended to
1000 by E. W. Weisstein (Nov. 17, 2003). 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,
... (OEIS A046066).
See also
Andrica's Conjecture, Euclid Number, PrimorialExplore with Wolfram|Alpha
References
Banderier, C. "Fortunate and Unfortunate Primes: Nearest Primes from a Prime Factorial." Dec. 18, 2000. https://web.archive.org/web/20070707232639/http://algo.inria.fr:80/banderier/Computations/prime_factorial.html.Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.Golomb, S. W. "The Evidence for Fortune's Conjecture." Math. Mag. 54, 209-210, 1981.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 7, 1994.Sloane, N. J. A. Sequences A005235/M2418, A006862/M2698, A035345, and A046066 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Fortunate PrimeCite this as:
Weisstein, Eric W. "Fortunate Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FortunatePrime.html