The first Fibonacci primes are (sequence A005478 in the OEIS): It is not known whether there are infinitely many Fibonacci primes.
With the indexing starting with F1 = F2 = 1, the first 37 indices n for which Fn is prime are (sequence A001605 in the OEIS): (Note that the actual values Fn rapidly become very large, so, for practicality, only the indices are listed.)
However, Fibonacci primes appear to become rarer as the index increases.
[3] The number of prime factors in the Fibonacci numbers with prime index are: As of September 2023[update], the largest known certain Fibonacci prime is F201107, with 42029 digits.
It was proved prime by Maia Karpovich in September 2023.
[4] The largest known probable Fibonacci prime is F10367321.
It was found by Ryan Propper in July 2024.
[2] It was proved by Nick MacKinnon that the only Fibonacci numbers that are also twin primes are 3, 5, and 13.
if and only if p is congruent to ±1 modulo 5, and p divides
(For p = 5, F5 = 5 so 5 divides F5) 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:[6] For n ≥ 3, Fn divides Fm if and only if n divides m.[7] If we suppose that m is a prime number p, and n is less than p, then it is clear that Fp cannot share any common divisors with the preceding Fibonacci numbers.
This means that Fp 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 Fn is to divide out the prime factors of all earlier Fibonacci numbers Fk for which k | n.[9] The exact quotients left over are prime factors that have not yet appeared.
If p and q are both primes, then all factors of Fpq are characteristic, except for those of Fp and Fq.
Therefore: The number of distinct prime factors of the Fibonacci numbers with a prime index is directly relevant to the counting function.
(sequence A080345 in the OEIS) For a prime p, the smallest index u > 0 such that Fu 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.[10] The rank of apparition divides the Pisano period π(p) and allows to determine all Fibonacci numbers divisible by p.[11] For the divisibility of Fibonacci numbers by powers of a prime,
as illustrated in the table below: The existence of Wall–Sun–Sun primes is conjectural.
The result is called the primitive part of
The primitive parts of the Fibonacci numbers are Any primes that divide
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 × 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
The number of primitive prime factors of
[13] 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,[14] a sequence which resembles the primes in some respects.