A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.
The first Fibonacci primes are :
It is not known whether there are infinitely many Fibonacci primes. With the indexing starting with , the first 37 indices n for which F<sub>n</sub> is prime are :
(Note that the actual values F<sub>n</sub> rapidly become very large, so, for practicality, only the indices are listed.)
In addition to these proven Fibonacci primes, several probable primes have been found:
Except for the case n = 4, all Fibonacci primes have a prime index, because if a divides b, then also divides (but not every prime index results in a Fibonacci prime). That is to say, the Fibonacci sequence is a divisibility sequence.
F<sub>p</sub> is prime for 8 of the first 10 primes p; the exceptions are F<sub>2</sub> = 1 and F<sub>19</sub> = 4181 = 37 × 113. However, Fibonacci primes appear to become rarer as the index increases. F<sub>p</sub> is prime for only 26 of the 1229 primes p smaller than 10,000. The number of prime factors in the Fibonacci numbers with prime index are:
, the largest known certain Fibonacci prime is F<sub>201107</sub>, with 42029 digits. It was proved prime by Maia Karpovich in September 2023. The largest known probable Fibonacci prime is F<sub>11964299</sub>. It was found by Ryan Propper in June 2025. It was proved by Nick MacKinnon that the only Fibonacci numbers that are also twin primes are 3, 5, and 13.
A prime divides if and only if p is congruent to ñ1 modulo 5, and p divides if and only if it is congruent to ñ2 modulo 5. (For p = 5, F<sub>5</sub> = 5 so 5 divides F<sub>5</sub>)
Fibonacci numbers that have a prime index p do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity:
For , F<sub>n</sub> divides F<sub>m</sub> if and only if n divides m.
If we suppose that m is a prime number p, and n is less than p, then it is clear that F<sub>p</sub> cannot share any common divisors with the preceding Fibonacci numbers.
This means that F<sub>p</sub> will always have characteristic factors or be a prime characteristic factor itself. The number of distinct prime factors of each Fibonacci number can be put into simple terms.
The first step in finding the characteristic quotient of any F<sub>n</sub> is to divide out the prime factors of all earlier Fibonacci numbers F<sub>k</sub> for which k | n.
The exact quotients left over are prime factors that have not yet appeared.
If p and q are both primes, then all factors of F<sub>pq</sub> are characteristic, except for those of F<sub>p</sub> and F<sub>q</sub>.
Therefore:
The number of distinct prime factors of the Fibonacci numbers with a prime index is directly relevant to the counting function.
For a prime p, the smallest index u > 0 such that F<sub>u</sub> is divisible by p is called the rank of apparition (sometimes called Fibonacci entry point) of p and denoted a(p). The rank of apparition a(p) is defined for every prime p. The rank of apparition divides the Pisano period ÃÂ(p) and allows to determine all Fibonacci numbers divisible by p.
For the divisibility of Fibonacci numbers by powers of a prime, and
In particular
A prime p â 2, 5 is called a FibonacciâÂÂWieferich prime or a WallâÂÂSunâÂÂSun prime if where
and is the Legendre symbol:
It is known that for p â 2, 5, a(p) is a divisor of:
For every prime p that is not a WallâÂÂSunâÂÂSun prime, as illustrated in the table below:
The existence of WallâÂÂSunâÂÂSun primes is conjectural.
Because , we can divide any Fibonacci number by the least common multiple of all where . The result is called the primitive part of . The primitive parts of the Fibonacci numbers are
Any primes that divide and not any of the s are called primitive prime factors of . The product of the primitive prime factors of the Fibonacci numbers are
The first case of more than one primitive prime factor is 4181 = 37 ÃÂ 113 for .
The primitive part has a non-primitive prime factor in some cases. The ratio between the two above sequences is
The natural numbers n for which has exactly one primitive prime factor are
For a prime p, p is in this sequence if and only if is a Fibonacci prime, and 2p is in this sequence if and only if is a Lucas prime (where is the th Lucas number). Moreover, 2<sup>n</sup> is in this sequence if and only if is a Lucas prime.
The number of primitive prime factors of are
The least primitive prime factors of are
It is conjectured that all the prime factors of are primitive when is a prime number.
Although it is not known whether there are infinitely many Fibonacci numbers which are prime, Melfi proved that there are infinitely many which are practical numbers, a sequence which resembles the primes in some respects.