Fibonacci periodicity and fibonacci frequency

Abstract

In this paper we introduce the notion of Fibonacci periodicity modulo n, denoting this period by the function b F(n). We note that b F(n) is an integral multiple of a fundamental frequency b f(n), where the ratio b F(n)= b f(n) is a power of 2 for a collection of observed values of n. It is demonstrated that if a; b are natural numbers with gcd(a; b) = 1, then b F(n) = lcmf b F(a); b F(b)g and thus that b F is a non-trivial example of a function which we refer to as radical. From observations it also seems clear that b F(ps+1) b F(ps) = p for primes p.