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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.

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

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).

For n≥3, Fn divides Fmm iff n divides m.

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

* Lucas number 