Periodicity of power Fibonacci sequences modulus a Fibonacci number

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 ${\mathcal{F}}^e(F_j)=(F_i^e(\mathrm{mod}{F}_j):i\ge 0)$. 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 ${\rho }_i\equiv F_i^e(\mathrm{mod}{F}_j)$ with $0\le {\rho }_i<F_j-1$ are obtained.