It is not known whether there are infinitely many Fibonacci primes. With the indexing starting with, the first 34 are Fn for the n values : In addition to these proven Fibonacci primes, there have been found probable primes for Except for the case n = 4, all Fibonacci primes have a prime index, because if a divides b, then also divides, but not every prime is the index of a Fibonacci prime. 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. The number of prime factors in the Fibonacci numbers with prime index are: , the largest known certain Fibonacci prime is F104911, with 21925 digits. It was proved prime by Mathew Steine and Bouk de Water in 2015. The largest known probable Fibonacci prime is F3340367. It was found by Henri Lifchitz in 2018. It was proved by Nick MacKinnon that the only Fibonacci numbers that are also members of the set of twin primes are 3, 5, and 13.
Divisibility of Fibonacci numbers
A prime divides if and only ifp is congruent to ±1 modulo 5, and p divides if and only if it is congruent to ±2 modulo 5. 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: which implies the infinitude of primes since is divisible by at least one prime for all. For, Fn divides Fm iff n divides m. If we suppose that m is a prime numberp, and n is less than p, then it is clear that Fp, cannot share any common divisors with the preceding Fibonacci numbers. 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 from 1 up. 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 4 special cases: and 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.
n
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
Fn
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
πn
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. 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. Therefore: The number of distinct prime factors of the Fibonacci numbers with a prime index is directly relevant to the counting function.
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
πp
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 of p and denoted a. The rank of apparition a is defined for every prime p. The rank of apparition divides the Pisano period π and allows to determine all Fibonacci numbers divisible by p. For the divisibility of Fibonacci numbers by powers of a prime, and In particular
Wall-Sun-Sun primes
A prime p ≠ 2, 5 is called a Fibonacci–Wieferich prime or a Wall-Sun-Sun prime if where in which is the Legendre symbol defined as: It is known that for p ≠ 2, 5, a is a divisor of: For every prime p that is not a Wall-Sun-Sun prime, 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
3
4
5
8
10
7
9
18
24
14
30
19
20
44
16
27
58
15
a
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.
The primitive part of the Fibonacci numbers are The product of the primitive prime factors of the Fibonacci numbers are The first case of more than one primitive prime factor is 4181 = 37 × 113 for. The primitive part has a non-primitive prime factor in some cases. The ratio between the two above sequences is The natural numbersn for which has exactly one primitive prime factor are If and only if a prime p is in this sequence, then is a Fibonacci prime, and if and only if 2p is in this sequence, then is a Lucas prime, and if and only if 2n is in this sequence, then is a Lucas prime. Number of primitive prime factors of are The least primitive prime factor of are
