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

Dishong, Alex; Renault, Marc S.

doi:10.2140/involve.2018.11.769

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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

DOI: 10.2140/involve.2018.11.769

Abstract

The Fibonacci sequence $F = 0, 1, 1, 2, 3, 5, 8, 13, \dots$ , when reduced modulo $m$ is periodic. For example, $F mod 4 = 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, \dots$ . The period of $F mod m$ is denoted by $π (m)$ , so $π (4) = 6$ . In this paper we present an algorithm that, given a period $k$ , produces all $m$ such that $π (m) = k$ . 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

Vol. 11, No. 5, 2018