A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime. The first Fibonacci primes are :

Known Fibonacci primes

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 : (Note that the actual values Fn 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 F_a also divides F_b (but not every prime index results in a Fibonacci prime). That is to say, the Fibonacci sequence is a divisibility sequence. Fp is prime for 8 of the first 10 primes p; the exceptions are F2 = 1 and F19 = 4181 = 37 × 113. However, Fibonacci primes appear to become rarer as the index increases. Fp 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 F201107, with 42029 digits. It was proved prime by Maia Karpovich in September 2023. The largest known probable Fibonacci prime is F10367321. It was found by Ryan Propper in July 2024. It was proved by Nick MacKinnon that the only Fibonacci numbers that are also twin primes are 3, 5, and 13.

Divisibility of Fibonacci numbers

A prime p divides F_{p-1} if and only if p is congruent to ±1 modulo 5, and p divides F_{p+1} if and only if it is congruent to ±2 modulo 5. (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: For n ≥ 3 , Fn divides Fm 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 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. 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.

Rank of apparition

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. The rank of apparition divides the Pisano period π(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

Wall–Sun–Sun primes

A prime p ≠ 2, 5 is called a Fibonacci–Wieferich prime or a Wall–Sun–Sun prime if where and is the Legendre symbol: It is known that for p ≠ 2, 5, a(p) is a divisor of: For every prime p that is not a Wall–Sun–Sun prime, as illustrated in the table below: The existence of Wall–Sun–Sun primes is conjectural.

Fibonacci primitive part

Because, we can divide any Fibonacci number F_n by the least common multiple of all F_d where d | n. The result is called the primitive part of F_n. The primitive parts of the Fibonacci numbers are Any primes that divide F_n and not any of the F_ds are called primitive prime factors of F_n. 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 F_{19}. 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 F_n has exactly one primitive prime factor are For a prime p, p is in this sequence if and only if F_p is a Fibonacci prime, and 2p is in this sequence if and only if L_p is a Lucas prime (where L_p is the pth Lucas number). Moreover, 2n is in this sequence if and only if L_{2^{n-1}} is a Lucas prime. The number of primitive prime factors of F_n are The least primitive prime factors of F_n are It is conjectured that all the prime factors of F_n are primitive when n is a prime number.

Fibonacci numbers in prime-like sequences

Although it is not known whether there are infinitely primes in the Fibonacci sequence, Melfi proved that there are infinitely many primes among practical numbers, a prime-like sequence.