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 , 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. (The values Fn rapidly become very large so only the indices, n, 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, 7789819, 10317107, 10367321, 11964299.PRP Top Records, Search for : F(n). Retrieved 2018-04-05.

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.

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.Sloane's , 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 F11964299. It was found by Ryan Propper in June 2025. It was proved by Nick MacKinnon that the only Fibonacci numbers that are also twin primes are 3, 5, and 13.N. MacKinnon, Problem 10844, Amer. Math. Monthly 109, (2002), p. 78

Divisibility of Fibonacci numbers 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, 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:Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000

For , Fn divides Fm if and only if n divides m.Wells 1986, p.65

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.

Fnk is a multiple of Fk for all values of n and k such that n ≥ 1 and k ≥ 1.The mathematical magic of Fibonacci numbers Factors of Fibonacci numbers 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: and 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 except for If k = 1, and n is an odd prime, then 1 | p and 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 Fn 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 πn 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 Fn is to divide out the prime factors of all earlier Fibonacci numbers Fk for which k | n.Jarden - Recurring sequences, Volume 1, Fibonacci quarterly, by Brother U. Alfred

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. 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 πp 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

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:

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(p2) 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.

Fibonacci primitive part 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 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 and not any of the s are called primitive prime factors of . 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 .

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 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 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, 2n is in this sequence if and only if is a Lucas prime.

The number of primitive prime factors of 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 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 are primitive when is a prime number.The mathematical magic of Fibonacci numbers Fibonacci Numbers and Primes

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

Fibonacci prime

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