Subsequences and Divisibility by Powers of the Fibonacci Numbers

TL;DR

This paper investigates the divisibility properties of a recursively defined Fibonacci-based sequence, establishing divisibility by powers of Fibonacci numbers and analyzing the sequence modulo Fibonacci numbers.

Contribution

It introduces a new recursive sequence based on Fibonacci numbers and proves its divisibility properties, extending understanding of Fibonacci divisibility patterns.

Findings

Proves that $F_n^{k+m-1}$ divides $G(k,n,m)$ for all positive integers.

Calculates the sequence's value modulo $F_n$, revealing its residue pattern.

Establishes a general divisibility framework for Fibonacci-based sequences.

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 $\frac{G(k,n,m)}{F_n^{k+m-1}}\pmod{F_n}$.