Fibonacci Prime - Divisibility of Fibonacci Numbers

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).

(This implies the infinitude of primes.)

For n ≥ 3, F_n divides F_m 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 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 (except for 1, 8 and 144) has at least one prime factor that has not been a factor of the preceding Fibonacci numbers.