A Fibonacci prime is a Fibonacci number that is prime. The first few Fibonacci primes are

2, 3, 5, 13, 89, 233, 1597, ...

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

F_p is prime for 8 out of the first 10 primes; the exceptions are F₂ = 1 and F₁₉ = 4181 = 37 x 113. However, Fibonacci primes become rarer as the index increases - F_{p is prime for only 25 of the 1,229 primes p below 10,000.Sloane's A005478, Sloane's A001605}

Currently, the largest known certain Fibonacci prime is F₈₁₈₃₉, with 17103 digitsNumber Theory Archives announcement by David Broadhurst and Bouk de Water; the largest known probable Fibonacci prime is F₆₀₄₇₁₁, with 126377 digits.PRP Records It is not known if there are infinitely many Fibonacci primes.

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(F_n, F_m) = F_GCD(n,m).Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000

For n≥3, F_n divides F_m iff n divides m.Wells 1986, p.65

The greatest common divisor between any two Fibonacci numbers, is equal to a Fibonacci number, with an index that is the greatest common divisor, of the index values held by the two Fibonacci numbers.

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 F_p, cannot share any common divisors with the preceding Fibonacci numbers.

GCD(F_p, F_n) = F_GCD(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