Fibonacci prime

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 F_n is prime are (sequence A001605 in the OEIS):

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 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:

n = 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367, 4740217, 6530879.[2]

Except for the case n = 4, all Fibonacci primes have a prime index, because if a divides b, then ${\displaystyle F_{a}}$ also divides ${\displaystyle F_{b}}$ (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 F₂ = 1 and F₁₉ = 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:

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, ... (sequence A080345 in the OEIS)

As of September 2023, the largest known certain Fibonacci prime is F₂₀₁₁₀₇, with 42029 digits. It was proved prime by Maia Karpovich in September 2023.[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 ${\displaystyle p}$ divides ${\displaystyle F_{p-1}}$ if and only if p is congruent to ±1 modulo 5, and p divides ${\displaystyle F_{p+1}}$ 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:[6]

${\displaystyle \gcd(F_{n},F_{m})=F_{\gcd(n,m)}.}$

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.

${\displaystyle \gcd(F_{p},F_{n})=F_{\gcd(p,n)}=F_{1}=1.}$

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.

• F_nk is a multiple of F_k for all values of n and k such that n ≥ 1 and k ≥ 1.[8] It's safe to say that F_nk will have "at least" the same number of distinct prime factors as F_k. All F_p will have no factors of F_k, but "at least" one new characteristic prime from Carmichael's theorem.

• Carmichael's Theorem applies to all Fibonacci numbers except four special cases: ${\displaystyle F_{1}=F_{2}=1,F_{6}=8}$ and ${\displaystyle F_{12}=144.}$ 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 π_n be the number of distinct prime factors of F_n. (sequence A022307 in the OEIS)

If k | n then ${\displaystyle \pi _{n}\geqslant \pi _{k}+1}$ except for ${\displaystyle \pi _{6}=\pi _{3}=1.}$

If k = 1, and n is an odd prime, then 1 | p and ${\displaystyle \pi _{p}\geqslant \pi _{1}+1=1.}$

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.

${\displaystyle {\begin{aligned}\gcd(F_{pq},F_{q})&=F_{\gcd(pq,q)}=F_{q}\\\gcd(F_{pq},F_{p})&=F_{\gcd(pq,p)}=F_{p}\end{aligned}}}$

Therefore:

${\displaystyle \pi _{pq}\geqslant {\begin{cases}\pi _{p}+\pi _{q}+1&p\neq q\\\pi _{p}+1&p=q\end{cases}}}$

The number of distinct prime factors of the Fibonacci numbers with a prime index is directly relevant to the counting function. (sequence A080345 in the OEIS)

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, ${\displaystyle p\geqslant 3,n\geqslant 2}$ and ${\displaystyle k\geqslant 0}$

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

In particular

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

A prime p ≠ 2, 5 is called a Fibonacci–Wieferich prime or a Wall–Sun–Sun prime if ${\displaystyle p^{2}\mid F_{q},}$ where

${\displaystyle q=p-\left({\frac {p}{5}}\right)}$

and ${\displaystyle \left({\tfrac {p}{5}}\right)}$ is the Legendre symbol:

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

It is known that for p ≠ 2, 5, a(p) is a divisor of:[12]

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

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

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

Because ${\displaystyle F_{a}|F_{ab}}$, we can divide any Fibonacci number ${\displaystyle F_{n}}$ by the least common multiple of all ${\displaystyle F_{d}}$ where ${\displaystyle d|n}$. The result is called the primitive part of ${\displaystyle F_{n}}$. 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, ... (sequence A061446 in the OEIS)

Any primes that divide ${\displaystyle F_{n}}$ and not any of the ${\displaystyle F_{d}}$s are called primitive prime factors of ${\displaystyle F_{n}}$. 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, ... (sequence A178763 in the OEIS)

The first case of more than one primitive prime factor is 4181 = 37 × 113 for ${\displaystyle F_{19}}$.

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, .... (sequence A178764 in the OEIS)

The natural numbers n for which ${\displaystyle F_{n}}$ 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, ... (sequence A152012 in the OEIS)

For a prime p, p is in this sequence if and only if ${\displaystyle F_{p}}$ is a Fibonacci prime, and 2p is in this sequence if and only if ${\displaystyle L_{p}}$ is a Lucas prime (where ${\displaystyle L_{p}}$ is the ${\displaystyle p}$th Lucas number). Moreover, 2ⁿ is in this sequence if and only if ${\displaystyle L_{2^{n-1}}}$ is a Lucas prime.

The number of primitive prime factors of ${\displaystyle F_{n}}$ 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 the OEIS)

The least primitive prime factors of ${\displaystyle F_{n}}$ 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 the OEIS)

It is conjectured that all the prime factors of ${\displaystyle F_{n}}$ are primitive when ${\displaystyle n}$ is a prime number.[13]

Although it is not known whether there are infinitely primes in the Fibonacci sequence, Melfi proved that there are infinitely many primes[14] among practical numbers, a prime-like sequence.

• Lucas number

• Weisstein, Eric W. "Fibonacci Prime". MathWorld.
• R. Knott Fibonacci primes
• Caldwell, Chris. Fibonacci number, Fibonacci prime, and Record Fibonacci primes at the Prime Pages
• Factorization of the first 300 Fibonacci numbers
• Factorization of Fibonacci and Lucas numbers Archived 2016-08-19 at the Wayback Machine
• Small parallel Haskell program to find probable Fibonacci primes at haskell.org