Tantalizing properties of subsequences of the Fibonacci sequence modulo 10

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.

Figures (12)

• Figure 1.1: The Pisano period $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, n}\right)_{n=0}^{59}$ with values equally spaced
• Figure 1.2: Subsequence diagram corresponding to $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 3 + 25 j}\right)_{j=0}^\infty$
• Figure 1.3: Subsequence diagram corresponding to $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 15 + 13 j}\right)_{j=0}^\infty$
• Figure 2.1: Subsequence diagrams for $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 3 + 12 j}\right)_{j=0}^\infty$, $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 3 + 25 j}\right)_{j=0}^\infty$, and $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 3 + 17 j}\right)_{j=0}^\infty$
• Figure 2.2: Subsequence diagrams for $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 0 + 9 j}\right)_{j=0}^\infty$, $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 0 + 21 j}\right)_{j=0}^\infty$, and $\left(\mathop{\mathrm{\mathcal{F}}}\nolimits_{10, 0 + 27 j}\right)_{j=0}^\infty$

Theorems & Definitions (64)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Example 2.4: The Fibonacci sequence modulo $10$
• Proposition 2.5
• proof
• Definition 2.6
• Definition 2.7
• Example 2.8
• Proposition 2.9