# Fibonacci prime

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

The first Fibonacci primes are (sequence A005478 in OEIS):

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

## Known Fibonacci primes

 Are there an infinite number of Fibonacci primes?

It is not known whether there are infinitely many Fibonacci primes. The first 33 are Fn for the n values (sequence A001605 in OEIS):

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.

In addition to these proven Fibonacci primes, there have been found probable primes for

n = 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353.[1]

Except for the case n = 4, all Fibonacci primes have a prime index, because if and only if a divides b, then $F_a$ also divides $F_b$, but not every prime is the index of a Fibonacci prime.

Fp is prime for 8 of the first 10 primes p; the exceptions are F2 = 1 and F19 = 4181 = 37 × 113. However, Fibonacci primes become rarer as the index increases. Fp is prime for only 26 of the 1,229 primes p below 10,000.[2]

As of August 2014, the largest known certain Fibonacci prime is F81839, with 17103 digits. It was proved prime by David Broadhurst and Bouk de Water in 2001.[3][4] The largest known probable Fibonacci prime is F2904353. It has 606974 digits and was found by Henri Lifchitz in 2014.[1] In completely different hand, Nick MacKinnon proved that the only Fibonacci numbers that are also members of the set of prime twins are 3, 5 and 13.[5]

## Divisibility of Fibonacci numbers

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(Fn, Fm) = FGCD(n,m).[6]

(This implies the infinitude of primes.)

For n ≥ 3, Fn divides Fm iff n divides m.[7]

If we suppose that m is a prime number p from the identity above, and n is less than p, then it is clear that Fp, cannot share any common divisors with the preceding Fibonacci numbers.

GCD(Fp, Fn) = FGCD(p,n) = F1 = 1

Carmichael's theorem states that every Fibonacci number (except for 1, 8 and 144) has at least one prime factor that has not been a factor of the preceding Fibonacci numbers.

If and only if a prime p congruent to 1 or 4 (mod 5), then p divides Fp-1, otherwise, p divides Fp+1. (The only exception is p = 5, if and only if p = 5, then p divides Fp)

## Fibonacci primitive part

The primitive part 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, ... (sequence A178763 in OEIS)

These natural number ns which the primitive part of $F_n$ is prime 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, ... (sequence A152012 in OEIS)

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, ... (sequence A086597 in OEIS)

The least primitive prime factor 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, ... (sequence A001578 in OEIS)

If and only if a prime p is in this sequence, then $F_p$ is a Fibonacci prime, and if and only if 2p is in this sequence, then $L_p$ is a Lucas prime (where $L_n$ is the Lucas sequence), and if and only if 2n is in this sequence, then $L_{2^{n-1}}$ is a Lucas prime.