A consideration of the Fibonacci sequence modulo m

TL;DR

This paper investigates properties of Fibonacci numbers modulo m, focusing on specific congruences and divisibility conditions, and establishes a characterization relating prime and composite moduli with Fibonacci divisibility patterns.

Contribution

It provides a new characterization of Fibonacci divisibility properties modulo m, linking prime and composite cases with specific divisibility criteria.

Findings

Identifies conditions for Fibonacci numbers modulo m with certain congruences.

Establishes a characterization of when m^2 divides Fibonacci numbers based on prime divisibility.

Shows that only m=6 or 12 satisfy particular Fibonacci divisibility conditions.

Abstract

In this paper, we study natural numbers $m$ with ${u_{z \pm 1} \equiv -1 \ (m)}$ for a $z \in \mathbb{N}$, where $u_n$ is the $n$th Fibonacci number. Furthermore, we want to show for $m \in \mathbb{N}\setminus\{0,1\}$: $[\forall p \ \text{prim}: p^2 \nmid u_{\gamma(p)}] \Leftrightarrow [m^2 \mid u_{\gamma(m)} \Rightarrow m \in \{6,12\}]$.