TANTALIZING PROPERTIES OF SUBSEQUENCES OF THE FIBONACCI SEQUENCE MODULO 10

Abstract

The Fibonacci sequence modulo $m$, which we denote ${({\mathcal{ℱ}}_{m,n})}_{n=0}^{\infty }$ where ${\mathcal{ℱ}}_{m,n}$ is the Fibonacci number $F_n$ 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 ${({\mathcal{ℱ}}_{m,n})}_{n=0}^{\infty }$ 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 ${({\mathcal{ℱ}}_{10,n})}_{n=0}^{\infty }$ starting from the term ${\mathcal{ℱ}}_{10,k}$ for some $0\le k\le 59$. More precisely we consider the subsequences ${({\mathcal{ℱ}}_{10,k+rj})}_{j=0}^{\infty }$, 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 ${({\mathcal{ℱ}}_{10,k+rj})}_{j=0}^{\infty }$ coincides exactly with the original parent sequence ${({\mathcal{ℱ}}_{10,n})}_{n=0}^{\infty }$ (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.