February 2024 TANTALIZING PROPERTIES OF SUBSEQUENCES OF THE FIBONACCI SEQUENCE MODULO 10
Dan Guyer, aBa Mbirika, Miko Scott
Rocky Mountain J. Math. 54(1): 179-206 (February 2024). DOI: 10.1216/rmj.2024.54.179

Abstract

The Fibonacci sequence modulo m, which we denote (m,n)n=0 where m,n is the Fibonacci number Fn modulo m, has been a well-studied object in mathematics since the seminal paper by D. D. Wall in 1960 exploring a myriad of properties related to the periods of these sequences. Since the time of Lagrange it has been known that (m,n)n=0 is periodic for each m. We examine this sequence when m=10, yielding a sequence of period length 60. In particular, we explore its subsequences composed of every r-th term of (10,n)n=0 starting from the term 10,k for some 0k59. More precisely we consider the subsequences (10,k+rj)j=0, which we show are themselves periodic and whose lengths divide 60. Many intriguing properties reveal themselves as we alter the k and r values. For example, for certain r values the corresponding subsequences surprisingly obey the Fibonacci recurrence relation, that is, any two consecutive subsequence terms sum to the next term modulo 10. Moreover, for all r values relatively prime to 60, the subsequence (10,k+rj)j=0 coincides exactly with the original parent sequence (10,n)n=0 (or a cyclic shift of it) running either forward or reverse. We demystify these phenomena and explore many other tantalizing properties of these subsequences. Lastly, we end with a few open questions, some of which ask whether results in this paper generalize to arbitrary moduli, but we provide evidence that m=10 may indeed be a very special case.

Version Information

The current online version of this article, posted on 9 October 2024, supersedes the version posted on 28 February 2024. A spelling error was corrected in the first "References" entry.

Citation

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Dan Guyer. aBa Mbirika. Miko Scott. "TANTALIZING PROPERTIES OF SUBSEQUENCES OF THE FIBONACCI SEQUENCE MODULO 10." Rocky Mountain J. Math. 54 (1) 179 - 206, February 2024. https://doi.org/10.1216/rmj.2024.54.179

Information

Received: 24 December 2021; Accepted: 12 November 2022; Published: February 2024
First available in Project Euclid: 28 February 2024

MathSciNet: MR4718513
Digital Object Identifier: 10.1216/rmj.2024.54.179

Subjects:
Primary: 11B39 , 11B50

Keywords: Fibonacci sequence , modular arithmetic , Pisano period , star polygon , subsequence

Rights: Copyright © 2024 Rocky Mountain Mathematics Consortium

Vol.54 • No. 1 • February 2024
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