The Fibonacci sequence under a modulus: computing all moduli that produce a given period

Abstract

The Fibonacci sequence $F=0,1,1,2,3,5,8,13,\dots$, when reduced modulo $m$ is periodic. For example, $Fmod4=0,1,1,2,3,1,0,1,1,2,\dots$. The period of $Fmodm$ is denoted by $\pi \left(m\right)$, so $\pi \left(4\right)=6$. In this paper we present an algorithm that, given a period $k$, produces all $m$ such that $\pi \left(m\right)=k$. For efficiency, the algorithm employs key ideas from a 1963 paper by John Vinson on the period of the Fibonacci sequence. We present output from the algorithm and discuss the results.