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(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

Famous quotes containing the word numbers:

    I’m not even thinking straight any more. Numbers buzz in my head like wasps.
    Kurt Neumann (1906–1958)