Fortunate Prime
المؤلف:
Gardner, M
المصدر:
"Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243
الجزء والصفحة:
...
23-9-2020
1040
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).
REFERENCES:
Banderier, C. "Fortunate and Unfortunate Primes: Nearest Primes from a Prime Factorial." Dec. 18, 2000. https://algo.inria.fr/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 A006862/M2698, A005235/M2418, A035345, and A046066 in "The On-Line Encyclopedia of Integer Sequences."
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