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 $(G(k,n,m))_{k\geq 1}$ by $G(1,n,m) = F^m_n$, and $G(k+1,n,m) = F_{nG(k,n,m)}$ for all $k\geq 1$. We show that $F_n^{k+m-1}\mid G(k,n,m)$ for all $k, m, n\in\mathbb N$. Then we calculate $\frac{G(k,n,m)}{F_n^{k+m-1}}\pmod{F_n}$.