Subsequences and Divisibility by Powers of the Fibonacci Numbers

Abstract

Let $F_n$ be the $n$th Fibonacci number. Let $m,n$ be positive integers. Define a sequence $(G(k,n,m){)}_{k\ge 1}$ by $G(1,n,m)=F_n^m$, and $G(k+1,n,m)=F_{nG(k,n,m)}$ for all $k\ge 1$. We show that $F_n^{k+m-1}\mid G(k,n,m)$ for all $k,m,n\in \mathbb{N}$. Then we calculate $\frac{G(k,n,m)}{F_n^{k+m-1}}(\mathrm{mod}{F}_n)$.

Publish in The Fibonacci Quarterly, May 2014