Fibonacci prime

A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.

The first Fibonacci primes are :


 * 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, ....

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 :
 * n = 3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107.

(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:
 * n = 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367, 4740217, 6530879.

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 &times; 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:
 * 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, 2, 3, 2, 2, 2, 2, 1, 1, 3, 4, 2, 4, 4, 2, 2, 3, 3, 2, 2, 4, 2, 4, 4, 2, 5, 3, 4, 3, 2, 3, 3, 4, 2, 2, 3, 4, 2, 4, 4, 4, 3, 2, 3, 5, 4, 2, 1, ...

, 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 F6530879. It was found by Ryan Propper in August 2022. 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:


 * $$\gcd(F_n, F_m) = F_{\gcd(n,m)}.$$

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.


 * $$\gcd(F_p, F_n) = F_{\gcd(p,n)} = F_1 = 1.$$

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.


 * Fnk is a multiple of Fk for all values of n and k such that n ≥ 1 and k ≥ 1. It's safe to say that Fnk will have "at least" the same number of distinct prime factors as Fk. All Fp will have no factors of Fk, but "at least" one new characteristic prime from Carmichael's theorem.


 * Carmichael's Theorem applies to all Fibonacci numbers except four special cases: $$F_1 =F_2 =1, F_6 = 8$$ and $$F_{12} = 144.$$ If we look at the prime factors of a Fibonacci number, there will be at least one of them that has never before appeared as a factor in any earlier Fibonacci number. Let πn be the number of distinct prime factors of Fn.
 * If k | n then $$\pi_n \geqslant \pi_k +1$$ except for $$\pi_6 = \pi_3 =1.$$
 * If k = 1, and n is an odd prime, then 1 | p and $$\pi_p \geqslant \pi_1 +1 = 1.$$

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.


 * $$\begin{align}

\gcd (F_{pq}, F_q ) &= F_{\gcd(pq, q)} = F_q \\ \gcd (F_{pq}, F_p ) &= F_{\gcd(pq, p)} = F_p \end{align}$$

Therefore:


 * $$\pi_{pq} \geqslant \begin{cases} \pi_p + \pi_q + 1 & p\neq q \\ \pi_p + 1 & p = q \end{cases}$$

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, $$p \geqslant 3, n \geqslant 2$$ and $$k \geqslant 0$$


 * $$p^n \mid F_{a(p)kp^{n-1}}.$$

In particular


 * $$p^2 \mid F_{a(p)p}.$$

Wall–Sun–Sun primes
A prime p ≠ 2, 5 is called a Fibonacci–Wieferich prime or a Wall–Sun–Sun prime if $$p^2 \mid F_q,$$ where


 * $$q = p - \left(\frac\right)$$

and $$\left(\tfrac\right)$$ is the Legendre symbol:


 * $$\left(\frac{p}{5}\right) = \begin{cases} 1 & p \equiv \pm 1 \bmod 5\\ -1 & p \equiv \pm 2 \bmod 5 \end{cases}$$

It is known that for p ≠ 2, 5, a(p) is a divisor of:


 * $$p-\left(\frac\right) = \begin{cases} p-1 & p \equiv \pm 1 \bmod 5\\ p+1 & p \equiv \pm 2 \bmod 5 \end{cases}$$

For every prime p that is not a Wall–Sun–Sun prime, $$a(p^2) = p a(p)$$ as illustrated in the table below:

The existence of Wall–Sun–Sun primes is conjectural.

Fibonacci primitive part
Because $$F_a | F_{ab}$$, 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
 * 1, 1, 2, 3, 5, 4, 13, 7, 17, 11, 89, 6, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 46, 15005, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 321, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441, ...

Any primes that divide $$F_n$$ and not any of the $$F_d$$s are called primitive prime factors of $$F_n$$. The product of the primitive prime factors of the Fibonacci numbers are
 * 1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 23, 3001, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 107, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441, 64079, 2971215073, 1103, 598364773, 15251, ...

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
 * 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, ....

The natural numbers n for which $$F_n$$ has exactly one primitive prime factor are
 * 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 45, 47, 48, 51, 52, 54, 56, 60, 62, 63, 65, 66, 72, 74, 75, 76, 82, 83, 93, 94, 98, 105, 106, 108, 111, 112, 119, 121, 122, 123, 124, 125, 131, 132, 135, 136, 137, 140, 142, 144, 145, ...

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 $$p$$th 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
 * 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 3, 2, 4, 1, 2, 2, 2, 2, 3, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, ...

The least primitive prime factors of $$F_n$$ are
 * 1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, 139, 2971215073, 1103, 97, 101, ...

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.