Periodicity of power Fibonacci sequences modulus a Fibonacci number

TL;DR

This paper investigates the periodicity of power Fibonacci sequences modulo Fibonacci numbers, extending known results for the case e=1 to higher powers, and explicitly computes their periods and residues.

Contribution

It generalizes the periodicity results of Fibonacci sequences modulo Fibonacci numbers to higher powers, providing explicit periods and residue values.

Findings

Sequences are periodic for all j,e≥1.

Explicit periods are computed for all cases.

Residues for e=1,2 are characterized.

Abstract

Let ${\mathcal F}=(F_i:i\ge 0)$ be the sequence of Fibonacci numbers, and $j$ and $e$ be non negative integers. We study the periodicity of the power Fibonacci sequences ${\mathcal F}^e(F_j)=(F_i^e\pmod{F_j}: i\ge 0)$. It is shown that for every $j,e\ge 1$ the sequence ${\mathcal F}^e(F_j)$ is periodic and its periodicity is computed. The result was previously known for ${\mathcal F}(F_j)$; that is, for $e=1$. For $e\in \{1, 2\}$, the values of the normalized residues $\rho_i\equiv F_i^e\pmod{F_j}$ with $0\le \rho_i<F_j-1$ are obtained.