Tantalizing properties of subsequences of the Fibonacci sequence modulo 10

TL;DR

This paper investigates the periodic properties and intriguing behaviors of Fibonacci sequence subsequences modulo 10, revealing surprising recurrence relations and symmetries, and providing a detailed analysis of their structure and relationships.

Contribution

It introduces a comprehensive analysis of Fibonacci subsequences modulo 10, uncovering new recurrence properties, symmetries, and conditions for sequence equivalence and shifts.

Findings

Subsequences are periodic with lengths dividing 60.

Certain subsequences obey the Fibonacci recurrence relation modulo 10.

Subsequences with r relatively prime to 60 match the original sequence or its reverse.

Abstract

The Fibonacci sequence modulo $m$, which we denote $\left(\mathcal{F}_{m,n}\right)_{n=0}^\infty$ 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 $\left(\mathcal{F}_{m,n}\right)_{n=0}^\infty$ 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 $r^{\mathrm{th}}$ term of $\left(\mathcal{F}_{10,n}\right)_{n=0}^\infty$ starting from the term $\mathcal{F}_{10,k}$ for some $0 \leq k \leq 59$. More precisely we consider the subsequences $\left(\mathcal{F}_{10,k+rj}\right)_{j=0}^\infty$, 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 $\left(\mathcal{F}_{10,k+rj}\right)_{j=0}^\infty$ coincides exactly with the original parent sequence $\left(\mathcal{F}_{10,n}\right)_{n=0}^\infty$ (or a cyclic shift of it) running either forward or reverse. We demystify this phenomena and explore many other tantalizing properties of these subsequences.