Elementary remarks about Pisano periods

TL;DR

This paper provides elementary proofs of known facts about Pisano periods and explores their connection to the order of roots of characteristic equations, using minimal ring theory background.

Contribution

It offers a simplified, elementary approach to understanding Pisano periods and extends techniques to linear recurrences with quadratic characteristic equations.

Findings

Elementary proofs of Pisano period properties

Link between Pisano periods and root orders established

Techniques applicable to broader classes of recurrences

Abstract

In this short note, we reprove in a very elementary way some known facts about Pisano periods as well as some considerations about the link between Pisano periods and the order of roots of the characteristic equation. The technics only requires a small background in ring theory (merely the definition of a commutative ring). The tools set here can be reused for all linear recurrences with quadratic non-constant characteristic equation.