Orbits of Second Order Linear Recurrences over Finite Fields

TL;DR

This paper investigates the structure and properties of orbits generated by second order linear recurrences over finite fields, focusing on orbit lengths, their frequencies, and conditions for roots to generate the entire multiplicative group.

Contribution

It introduces new insights into the orbit structure of these recurrences and links their properties to the algebraic characteristics of the associated matrices and polynomials.

Findings

Analysis of orbit lengths and their distribution

Conditions for roots to generate the entire multiplicative group

Connections between matrix order and recurrence properties

Abstract

Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the set $\mathbb{F}_q \times \mathbb{F}_q$. In particular, we study certain relationships between the lengths of the non-trivial orbits of $G$, and their frequency of occurrence. This is done in part by investigating the order of elements of a product in an abelian group when the product has prime power order. For $q$ a prime and $b=1$, the orbits correspond to Fibonacci type linear recurrences modulo $q$ for different initial conditions. We also derive certain conditions under which the roots of the characteristic polynomial of $Q$ are generators of $\mathbb{F}_q^\times$. Examples are included to illustrate the theory.