Fibonacci prime

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Prime number in the Fibonacci sequence

title: "Fibonacci prime" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["classes-of-prime-numbers", "fibonacci-numbers", "unsolved-problems-in-number-theory"] description: "Prime number in the Fibonacci sequence" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Fibonacci_prime" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Prime number in the Fibonacci sequence ::

::data[format=table title="Infobox integer sequence"]

FieldValue
terms_number17
con_numberInfinite
first_terms2, 3, 5, 13, 89, 233
largest_known_termF11964299
OEISA001605
OEIS_nameIndices of prime Fibonacci numbers
::

| terms_number = 17 | con_number = Infinite | first_terms = 2, 3, 5, 13, 89, 233 | largest_known_term = F11964299 | OEIS = A001605 | OEIS_name = Indices 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 :

: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 F**n 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 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, 7789819, 10317107, 10367321, 11964299.

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.

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 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 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.

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 F**p 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.

  • F**nk is a multiple of F**k 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 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: 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. ::data[format=table] | n | Fn | πn | |---|---|---| | 0 | 1 | 2 | | 0 | 1 | 1 | | 0 | 0 | 0 | ::

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. ::data[format=table]

pπp
23
01
::

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

Main article: Wall–Sun–Sun prime

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:

::data[format=table]

pa(p)a(p2)
235
345
61225
::

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_ds 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 pth 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 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.

References

References

  1. "Fibonacci Prime".
  2. [http://www.primenumbers.net/prptop/searchform.php?form=F%28n%29&action=Search PRP Top Records, Search for : F(n)]. Retrieved 2018-04-05.
  3. Sloane's {{OEIS2C. A005478, {{OEIS2C. A001605
  4. "The Top Twenty: Fibonacci Number".
  5. Luhn, Norman. (28 June 2025). "Fibonacci (probable) Primes".
  6. N. MacKinnon, Problem 10844, Amer. Math. Monthly 109, (2002), p. 78
  7. [[Paulo Ribenboim]], ''My Numbers, My Friends'', Springer-Verlag 2000
  8. Wells 1986, p.65
  9. The mathematical magic of Fibonacci numbers [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section2 Factors of Fibonacci numbers]
  10. Jarden - Recurring sequences, Volume 1, Fibonacci quarterly, by Brother U. Alfred
  11. {{OEIS. A001602
  12. John Vinson. (1963). "The Relation of the Period Modulo ''m'' to the Rank of Apparition of ''m'' in the Fibonacci Sequence". Fibonacci Quarterly.
  13. Steven Vajda. "Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications".
  14. The mathematical magic of Fibonacci numbers [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section5 Fibonacci Numbers and Primes]
  15. Giuseppe Melfi. (1995). "A survey on practical numbers". Rend. Sem. Mat. Torino.

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