On the number of residues of linear recurrences

TL;DR

This paper investigates the set of possible numbers of distinct residues modulo M that linear recurrences with a given characteristic polynomial can have, especially when the polynomial is divisible by a quadratic with roots related to Lehmer sequences.

Contribution

It extends previous work by analyzing cases where the polynomial is divisible by a quadratic with roots satisfying specific algebraic properties, linking the problem to primitive divisors of Lehmer sequences.

Findings

For roots with product -1, all integers m ≥ 7 are in the set, except 10 and multiples of 4.

The problem relates to the existence of primitive divisors in Lehmer sequences.

Provides new insights into the structure of residues of linear recurrences with special characteristic polynomials.

Abstract

For every nonconstant monic polynomial $g \in \mathbb{Z}[X]$, let $\mathfrak{M}(g)$ be the set of positive integers $m$ for which there exist an integer linear recurrence $(s_n)_{n \geq 0}$ having characteristic polynomial $g$ and a positive integer $M$ such that $(s_n)_{n \geq 0}$ has exactly $m$ distinct residues modulo $M$. Dubickas and Novikas proved that $\mathfrak{M}(X^2 - X - 1) = \mathbb{N}$. We study $\mathfrak{M}(g)$ in the case in which $g$ is divisible by a monic quadratic polynomial $f \in \mathbb{Z}[X]$ with roots $\alpha,\beta$ such that $\alpha\beta = \pm 1$ and $\alpha / \beta$ is not a root of unity. We show that this problem is related to the existence of special primitive divisors of certain Lehmer sequences, and we deduce some consequences on $\mathfrak{M}(g)$. In particular, for $\alpha\beta = -1$, we prove that $m \in \mathfrak{M}(g)$ for every integer $m \geq 7$ with $m \neq 10$ and $4 \nmid m$.