Abstract
The Fibonacci sequence modulo , which we denote where is the Fibonacci number modulo , 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 is periodic for each . We examine this sequence when , yielding a sequence of period length 60. In particular, we explore its subsequences composed of every -th term of starting from the term for some . More precisely we consider the subsequences , which we show are themselves periodic and whose lengths divide 60. Many intriguing properties reveal themselves as we alter the and values. For example, for certain 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 values relatively prime to 60, the subsequence coincides exactly with the original parent sequence (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 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
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
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