No. of known terms  51 

Conjectured no. of terms  Infinite^{[1]} 
First terms  2, 3, 5, 13, 89, 233 
Largest known term  F_{6530879} 
OEIS index 

A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.
The first Fibonacci primes are (sequence A005478 in the OEIS):
Are there an infinite number of Fibonacci primes?
It is not known whether there are infinitely many Fibonacci primes. With the indexing starting with F_{1} = F_{2} = 1, the first 34 indices n for which F_{n} is prime are (sequence A001605 in the OEIS):
(Note that the actual values F_{n} 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_{p} is prime for 8 of the first 10 primes p; the exceptions are F_{2} = 1 and F_{19} = 4181 = 37 × 113. However, Fibonacci primes appear to become rarer as the index increases. F_{p} is prime for only 26 of the 1,229 primes p below 10,000.^{[3]} The number of prime factors in the Fibonacci numbers with prime index are:
As of August 2022^{[update]}, the largest known certain Fibonacci prime is F_{148091}, with 30949 digits. It was proved prime by Laurent Facq et al. in September 2021.^{[4]} The largest known probable Fibonacci prime is F_{6530879}. It was found by Ryan Propper in August 2022.^{[2]} It was proved by Nick MacKinnon that the only Fibonacci numbers that are also twin primes are 3, 5, and 13.^{[5]}
A prime divides if and only if p is congruent to ±1 modulo 5, and p divides if and only if it is congruent to ±2 modulo 5. (For p = 5, F_{5} = 5 so 5 divides F_{5})
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, F_{n} divides F_{m} 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 F_{p} cannot share any common divisors with the preceding Fibonacci numbers.
This means that F_{p} 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.
n  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25 

F_{n}  0  1  1  2  3  5  8  13  21  34  55  89  144  233  377  610  987  1597  2584  4181  6765  10946  17711  28657  46368  75025 
π_{n}  0  0  0  1  1  1  1  1  2  2  2  1  2  1  2  3  3  1  3  2  4  3  2  1  4  2 
The first step in finding the characteristic quotient of any F_{n} is to divide out the prime factors of all earlier Fibonacci numbers F_{k} 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 F_{pq} are characteristic, except for those of F_{p} and F_{q}.
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)
p  2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97 

π_{p}  0  1  1  1  1  1  1  2  1  1  2  3  2  1  1  2  2  2  3  2  2  2  1  2  4 
For a prime p, the smallest index u > 0 such that F_{u} 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, and
In particular
Main article: Wall–Sun–Sun prime 
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:^{[12]}
For every prime p that is not a Wall–Sun–Sun prime, as illustrated in the table below:
p  2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61 

a(p)  3  4  5  8  10  7  9  18  24  14  30  19  20  44  16  27  58  15 
a(p^{2})  6  12  25  56  110  91  153  342  552  406  930  703  820  1892  752  1431  3422  915 
The existence of Wall–Sun–Sun 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 × 113 for .
The primitive part has a nonprimitive 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^{n} 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.^{[13]}
Although it is not known weather there are infinitely primes in the Fibonacci sequence, Melfi proved that there are infinitely many primes^{[14]} among practical numbers, a primelike sequence.