Fibonacci prime

Fibonacci prime
No. of known terms	51
Conjectured no. of terms	Infinite
First terms	2, 3, 5, 13, 89, 233
Largest known term	F3340367
OEIS index	A001605; 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 (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 are F_n for the n values (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.

In addition to these proven Fibonacci primes, there have been found probable primes for

n = 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367.^[2]

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).

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 March 2017, the largest known certain Fibonacci prime is F₁₀₄₉₁₁, with 21925 digits. It was proved prime by Mathew Steine and Bouk de Water in 2015.^[4] The largest known probable Fibonacci prime is F_3340367. It was found by Henri Lifchitz in 2018.^[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 $p$ 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:^[6]

\gcd(F_{n},F_{m})=F_{\gcd(n,m)},

which implies the infinitude of primes since $F_{p}$ is divisible by at least one prime for all $p>2$ .

For $n \geq 3$ , 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.

\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: $F_{1}=F_{2}=1,F_{6}=8$ 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 π_n be the number of distinct prime factors of F_n. (sequence A022307 in the OEIS)

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.

n	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
F_n	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
π_n	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 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.

{\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:

\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	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
π_p	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 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, $p\geqslant 3,n\geqslant 2$ 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[]

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 {p}{5}}\right)

in which $\left({\tfrac {p}{5}}\right)$ is the Legendre symbol defined as:

\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]

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, $a(p^{2})=pa(p)$ 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[]

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 $F_{19}$ .

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 $F_{n}$ 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 $F_{p}$ 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 $p$ th 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, ... (sequence A086597 in the OEIS)

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, ... (sequence A001578 in the OEIS)

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

References[]

^ http://mathworld.wolfram.com/FibonacciPrime.html
^ Jump up to: ^a ^b PRP Top Records, Search for : F(n). Retrieved 2018-04-05.
^ Sloane's OEIS: A005478, OEIS: A001605
^ Chris Caldwell, The Prime Database: U(104911) from the Prime Pages. Status: Fibonacci number, Elliptic Curve Primality Proof. Retrieved 2018-04-05.
^ N. MacKinnon, Problem 10844, Amer. Math. Monthly 109, (2002), p. 78
^ Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000
^ Wells 1986, p.65
^ The mathematical magic of Fibonacci numbers Factors of Fibonacci numbers
^ Jarden - Recurring sequences, Volume 1, Fibonacci quarterly, by Brother U. Alfred
^ (sequence A001602 in the OEIS)
^ John Vinson (1963). "The Relation of the Period Modulo m to the Rank of Apparition of m in the Fibonacci Sequence" (PDF). Fibonacci Quarterly. 1: 37–45.
^ Steven Vajda. Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications. Dover Books on Mathematics.
^ The mathematical magic of Fibonacci numbers Fibonacci Numbers and Primes

External links[]

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
Small parallel Haskell program to find probable Fibonacci primes at haskell.org

[1] ttp://mathworld.wolfram.com/FibonacciPrime.html

[prptop-2] Jump up to: ^a ^b PRP Top Records, Search for : F(n). Retrieved 2018-04-05.

[3] Sloane's OEIS: A005478, OEIS: A001605

[4] Chris Caldwell, The Prime Database: U(104911) from the Prime Pages. Status: Fibonacci number, Elliptic Curve Primality Proof. Retrieved 2018-04-05.

[5] N. MacKinnon, Problem 10844, Amer. Math. Monthly 109, (2002), p. 78

[6] Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000

[7] Wells 1986, p.65

[8] The mathematical magic of Fibonacci numbers Factors of Fibonacci numbers

[9] Jarden - Recurring sequences, Volume 1, Fibonacci quarterly, by Brother U. Alfred

[10] (sequence A001602 in the OEIS)

[11] John Vinson (1963). "The Relation of the Period Modulo m to the Rank of Apparition of m in the Fibonacci Sequence" (PDF). Fibonacci Quarterly. 1: 37–45.

[12] Steven Vajda. Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications. Dover Books on Mathematics.

[13] The mathematical magic of Fibonacci numbers Fibonacci Numbers and Primes

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