# 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:

*u*

_{1}=

*u*

_{2}= 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 *u*_{2}=1) or a prime. This
however is not sufficient!

The known Fibonacci primes
are *u*_{n} with

These have been tested by Dubner and Keller ton= 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.

*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, *u*_{7}=13, and if we
reverse these digits we get 31 which is also prime
(so *u*_{7} 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. MontgomeryandR. Silverman, "Tables of Fibonacci and Lucas factorizations,"Math. Comp.,50(1988) 251--260.MR 89h:11002- BMS88
J. Brillhart,P. L. MontgomeryandR. 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. DubnerandW. Keller, "New Fibonacci and Lucas primes,"Math. Comp.,68:225 (1999) 417--427, S1--S12.MR 99c:11008[Probable primality ofF,L,F*andL*tested fornup to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]