AFibonacci prime is aFibonacci number that isprime, a type of integer sequence prime.

The first Fibonacci primes are (sequenceA005478 in theOEIS):

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

It is not known whether there areinfinitely many Fibonacci primes. With the indexing starting withF₁ =F₂ = 1, the first 37 indicesn for whichF_n is prime are (sequenceA001605 in theOEIS):

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 valuesF_n rapidly become very large, so, for practicality, only the indices are listed.)

In addition to these proven Fibonacci primes, severalprobable 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.^[2]

Except for the casen = 4, all Fibonacci primes have a prime index, because ifadividesb, 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 adivisibility sequence.

F_p is prime for 8 of the first 10 primesp; the exceptions areF₂ = 1 andF₁₉ = 4181 = 37 × 113. However, Fibonacci primes appear to become rarer as the index increases.F_p is prime for only 26 of the 1229 primesp smaller than 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, ... (sequenceA080345 in theOEIS)

As of September 2023^[update], the largest known certain Fibonacci prime isF₂₀₁₁₀₇, with 42029 digits. It was proved prime by Maia Karpovich in September 2023.^[4][5] The largest known probable Fibonacci prime isF_11964299. It was found by Ryan Propper in June 2025.^[2]It was proved by Nick MacKinnon that the only Fibonacci numbers that are alsotwin primes are 3, 5, and 13.^[6]

A prime${\displaystyle p}$ divides${\displaystyle F_{p-1}}$if and only ifp iscongruent to ±1 modulo 5, andp divides${\displaystyle F_{p+1}}$ if and only if it is congruent to ±2 modulo 5. (Forp = 5,F₅ = 5 so 5 dividesF₅)

Fibonacci numbers that have a prime indexp do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity:^[7]

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

Forn ≥ 3,F_n dividesF_m if and only ifn dividesm.^[8]

If we suppose thatm is a prime numberp, andn is less thanp, then it is clear thatF_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 thatF_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 ofF_k for all values ofn andk such thatn ≥ 1 andk ≥ 1.^[9] It's safe to say thatF_nk will have "at least" the same number of distinct prime factors asF_k. AllF_p will have no factors ofF_k, but "at least" one new characteristic prime fromCarmichael'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. Letπ_n be the number of distinct prime factors ofF_n. (sequenceA022307 in theOEIS)

Ifk |n then${\displaystyle {\pi }_n⩾{\pi }_k+1}$ except for${\displaystyle {\pi }_6={\pi }_3=1.}$

Ifk = 1, andn is anodd prime, then 1 |p and${\displaystyle {\pi }_p⩾{\pi }_1+1=1.}$

The first step in finding the characteristic quotient of anyF_n is to divide out the prime factors of all earlier Fibonacci numbersF_k for whichk |n.^[10]

The exact quotients left over are prime factors that have not yet appeared.

Ifp andq are both primes, then all factors ofF_pq are characteristic, except for those ofF_p andF_q.

Therefore:

The number of distinct prime factors of the Fibonacci numbers with a prime index is directly relevant to the counting function. (sequenceA080345 in theOEIS)

For a primep, the smallest indexu > 0 such thatF_u is divisible byp is called therank of apparition (sometimes calledFibonacci entry point) ofp and denoteda(p). The rank of apparitiona(p) is defined for every primep.^[11] The rank of apparition divides thePisano period π(p) and allows to determine all Fibonacci numbers divisible byp.^[12]

For the divisibility of Fibonacci numbers by powers of a prime,${\displaystyle p⩾3,n⩾2}$ and${\displaystyle k⩾0}$

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

In particular

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

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

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

and${\displaystyle \left(\frac{p}{5}\right)}$ is theLegendre symbol:

It is known that forp ≠ 2, 5,a(p) is a divisor of:^[13]

For every primep 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 isconjectural.

Because${\displaystyle F_a|F_{ab}}$, we can divide any Fibonacci number${\displaystyle F_n}$ by theleast common multiple of all${\displaystyle F_d}$ where${\displaystyle d|n}$. The result is called theprimitive 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, ... (sequenceA061446 in theOEIS)

Any primes that divide${\displaystyle F_n}$ and not any of the${\displaystyle F_d}$s are calledprimitive 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, ... (sequenceA178763 in theOEIS)

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, .... (sequenceA178764 in theOEIS)

Thenatural numbersn 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, ... (sequenceA152012 in theOEIS)

For a primep,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 aLucas prime (where${\displaystyle L_p}$ is the${\displaystyle p}$thLucas 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, ... (sequenceA086597 in theOEIS)

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, ... (sequenceA001578 in theOEIS)

It is conjectured that all the prime factors of${\displaystyle F_n}$ are primitive when${\displaystyle n}$ is a prime number.^[14]

Although it is not known whether there are infinitely many Fibonacci numbers which are prime,Melfi proved that there are infinitely many which arepractical numbers,^[15] a sequence which resembles the primes in some respects.

• Lucas number

• Weisstein, Eric W."Fibonacci Prime".MathWorld.
• R. KnottFibonacci primes
• Caldwell, Chris.Fibonacci number,Fibonacci prime, andRecord Fibonacci primes at thePrime Pages
• Factorization of the first 300 Fibonacci numbers
• Factorization of Fibonacci and Lucas numbersArchived 2016-08-19 at theWayback Machine
• Small parallel Haskell program to find probable Fibonacci primes athaskell.org
• C++ program to find prime numbers in a fibonacci series

• Classes of prime numbers
• Fibonacci numbers
• Unsolved problems in number theory

• Articles with short description
• Short description matches Wikidata
• Articles containing potentially dated statements from September 2023
• All articles containing potentially dated statements
• Webarchive template wayback links