Vol. 11, No. 5, 2018

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

Alex Dishong and Marc S. Renault

Vol. 11 (2018), No. 5, 769–774
Abstract

The Fibonacci sequence $F=0,1,1,2,3,5,8,13,\dots \phantom{\rule{0.3em}{0ex}}$, when reduced modulo $m$ is periodic. For example, $F\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}4=0,1,1,2,3,1,0,1,1,2,\dots \phantom{\rule{0.3em}{0ex}}$. The period of $F\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}m$ 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.

Keywords
Fibonacci sequence, period, algorithm
Mathematical Subject Classification 2010
Primary: 11B39, 11B50
Secondary: 11Y55