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Fibonacci prime (another Prime Pages' Glossary entries) |
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A Fibonacci prime, as you should easily guess, is
a Fibonacci number that is prime. Recall that the
Fibonacci numbers can be defined as follows:
u1 = u2 = 1 and
un+1 = un + un-1 (n > 2).
It is easy to show that un divides unm (see primitive part of a Fibonacci number), so for un to be a prime, the subscript must either be 4 (because u2=1) or a prime. This however is not sufficient! The known Fibonacci primes are un with n = 3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, and 81839.These have been tested by Dubner and Keller to n=100,000 [DK99]. Others have extended this search, most notably Henri Lifchitz who has now reached past n=434,000. Besides the primes above, there are also the probable-primes when n = 37511, 50833, 104911 [Bouk de Water], 130021 [D. Fox], 148091 [T. D. Noe] and 201107, 397379, 433781 [H. Lifchitz] It seems likely that there are infinitely many Fibonacci primes, but this has yet to be proven. However, it is relatively easy to show that for n > 4, un+1 is never prime. A few folks have asked "what if we reverse the digits of the Fibonacci numbers?" For example, u7=13, and if we reverse these digits we get 31 which is also prime (so u7 is a reversable prime). The first Fibonacci numbers which form primes when their digits are reversed are those with: n = 3, 4, 5, 7, 9, 14, 17, 21, 25, 26, 65, 98, 175, 191, 382, 497, 653, 1577, 1942, 1958, 2405, 4246, 4878, 5367
See Also: LucasNumber Related pages (outside of this work)
References:
Chris K. Caldwell © 1999-2015 (all rights reserved)
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