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

The Fibonacci sequence F = 0,1,1,2,3,5,8,13,, when reduced modulo m is periodic. For example, F mod 4 = 0,1,1,2,3,1,0,1,1,2,. 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.

Fibonacci sequence, period, algorithm
Mathematical Subject Classification 2010
Primary: 11B39, 11B50
Secondary: 11Y55
Received: 2 June 2016
Accepted: 9 September 2017
Published: 2 April 2018

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