On the number of residues of linear recurrences

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 \((s_n)_{n \ge 0}\) having characteristic polynomial g and a positive integer M such that \((s_n)_{n \ge 0}\) 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 \(m \ge 7\) with \(m \ne 10\) and \(4 \not \mid m\).