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# Fibonacci prime

A Fibonacci prime is a Fibonacci number that is prime.

The first Fibonacci primes are (sequence A005478 in OEIS):

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

Known Fibonacci primes

It is not known if there are infinitely many Fibonacci primes. The first 33 are Fn for the n values A001605:

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

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

n = 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711

Except for the case n = 4, if Fn is prime then n is prime. The converse is false, however.

Fp is prime for 8 out of the first 10 primes; the exceptions are F2 = 1 and F19 = 4181 = 37 × 113. However, Fibonacci primes become rarer as the index increases - Fp is prime for only 25 of the 1,229 primes p below 10,000.[1]

As of 2006, the largest known certain Fibonacci prime is F81839, with 17103 digits.[2] The largest known probable Fibonacci prime is F604711. It has 126377 digits and was found by Henri Lifchitz in 2005.[3]

Divisibility of Fibonacci numbers

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(Fn, Fm) = FGCD(n,m).[4]

For n≥3, Fn divides Fmm iff n divides m.[5]

If we suppose that m, is a prime number p from the identity above, and n is less than p, then it is clear that Fp, cannot share any common divisors with the preceding Fibonacci numbers.

GCD(Fp, Fn) = FGCD(p,n) = F1 = 1

Carmichael's theorem states that every Fibonacci number (with a small set of exceptions) has at least one unique prime factor that has not been a factor of the preceding Fibonacci numbers.

References

1. ^ Sloane's A005478, Sloane's A001605

3. ^ PRP Records

4. ^ Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000

5. ^ Wells 1986, p.65