Fibonacci Prime - Known Fibonacci Primes

Known Fibonacci Primes

It is not known if there are infinitely many Fibonacci primes. The first 33 are F_n for the n values (sequence A001605 in OEIS):

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, 37511, 50833, 81839.

In addition to these proven Fibonacci primes, there have been found probable primes for

n = 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721.

Except for the case n = 4, all Fibonacci primes have a prime index, but not every prime is the index of a Fibonacci prime.

F_p is prime for 8 of the first 10 primes p; the exceptions are F₂ = 1 and F₁₉ = 4181 = 37 × 113. However, Fibonacci primes become rarer as the index increases. F_p is prime for only 25 of the 1,229 primes p below 10,000.

As of November 2009, the largest known certain Fibonacci prime is F₈₁₈₃₉, with 17103 digits. It was proved prime by David Broadhurst and Bouk de Water in 2001. The largest known probable Fibonacci prime is F_1968721. It has 411439 digits and was found by Henri Lifchitz in 2009.

By contrast, Nick MacKinnon proved that the only Fibonacci numbers that are also members of the set of prime twins are 3, 5 and 13.