A consideration of the Fibonacci sequence modulo m

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 $m \in \mathbb{N}\setminus\{0,1\}$: $[\forall p \ \text{prim}: p^2 \nmid u_{\gamma(p)}] \Leftrightarrow [m^2 \mid u_{\gamma(m)} \Rightarrow m \in \{6,12\}]$.