Fibonacci prime

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 u_n+1 = u_n + u_n-1 (n > 2).

It is easy to show that u_n divides u_nm (see primitive part of a Fibonacci number), so for u_n 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 u_n 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, u_n+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)

• Possible Fibonacci Primes by Dudley Fox
• Proof that U(81839) is prime by David Broadhurst and Bouk de Water

References:

BMS1988

J. Brillhart, P. Montgomery and R. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260.  MR 89h:11002

BMS88

J. Brillhart, P. L. Montgomery and R. D. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260, S1--S15.  MR 89h:11002 [See also [DK99].]

Brillhart1999

J. Brillhart, "Note on Fibonacci primality testing," Fibonacci Quart., 36:3 (1998) 222--228.  MR1627388

DK99

H. Dubner and W. Keller, "New Fibonacci and Lucas primes," Math. Comp., 68:225 (1999) 417--427, S1--S12.  MR 99c:11008 [Probable primality of F, L, F* and L* tested for n up to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]