# Fibonacci prime

Short description: Prime number in the Fibonacci sequence
No. of known terms 51 Infinite[1] 2, 3, 5, 13, 89, 233 F6530879 A001605Indices of prime Fibonacci numbers

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):

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

## Known Fibonacci primes

 Unsolved problem in mathematics:Are there an infinite number of Fibonacci primes?(more unsolved problems in mathematics)

It is not known whether there are infinitely many Fibonacci primes. With the indexing starting with F1 = F2 = 1, the first 34 indices n for which Fn 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.

(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 = 201107, 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.

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 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 August 2022), the largest known certain Fibonacci prime is F148091, with 30949 digits. It was proved prime by Laurent Facq et al. in September 2021.[4] The largest known probable Fibonacci prime is F6530879. 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]

## Divisibility of Fibonacci numbers

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, Fn divides Fm 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 Fp 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 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.[8] 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: $\displaystyle{ F_1 =F_2 =1, F_6 = 8 }$ 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 πn be the number of distinct prime factors of Fn. (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 $\displaystyle{ \pi_p \geqslant \pi_1 +1 = 1. }$
 n Fn πn 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 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 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.[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 Fpq are characteristic, except for those of Fp and Fq.

\displaystyle{ \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:

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

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)

 p πp 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 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.[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

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

## Wall–Sun–Sun primes

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, $\displaystyle{ a(p^2) = p a(p) }$ as illustrated in the table below:

 p a(p) a(p2) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 3 4 5 8 10 7 9 18 24 14 30 19 20 44 16 27 58 15 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

The primitive part 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)

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]

## References

1. PRP Top Records, Search for : F(n). Retrieved 2018-04-05.
2. Sloane's ,
3. N. MacKinnon, Problem 10844, Amer. Math. Monthly 109, (2002), p. 78
4. Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000
5. Wells 1986, p.65
6. The mathematical magic of Fibonacci numbers Factors of Fibonacci numbers
7. Jarden - Recurring sequences, Volume 1, Fibonacci quarterly, by Brother U. Alfred
8. (sequence A001602 in the OEIS)
9. John Vinson (1963). "The Relation of the Period Modulo m to the Rank of Apparition of m in the Fibonacci Sequence". Fibonacci Quarterly 1: 37–45.
10. Steven Vajda. Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications. Dover Books on Mathematics.
11. The mathematical magic of Fibonacci numbers Fibonacci Numbers and Primes