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]}
which implies the infinitude of primes since is divisible by at least one prime for all .
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.
The primitive part of the Fibonacci numbers are
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]}