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).
(This implies the infinitude of primes.)
For n ≥ 3, Fn divides Fm 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 (except for 1, 8 and 144) has at least one prime factor that has not been a factor of the preceding Fibonacci numbers.
Read more about this topic: Fibonacci Prime
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