#### 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
##### Milestones
Received: 2 June 2016
Accepted: 9 September 2017
Published: 2 April 2018

Communicated by Kenneth S. Berenhaut
##### Authors
 Alex Dishong Department of Mathematical Sciences University of Delaware Newark, DE United States Marc S. Renault Mathematics Department Shippensburg University Shippensburg, PA United States