On the number of residues of certain second-order linear recurrences

Abstract

For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f) \geq 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*} \mathcal{R}(f) := \big\{\rho(\mathbf{x}; m) : \mathbf{x} \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big\} , \end{equation*} where $\rho(\mathbf{x}; m)$ is the number of distinct residues of $\mathbf{x}$ modulo $m$. Dubickas and Novikas proved that $\mathcal{R}(X^2 - X - 1) = \mathbb{Z}^+$. We generalize this result by showing that $\mathcal{R}(X^2 - a_1 X - 1) = \mathbb{Z}^+$ for every nonzero integer $a_1$. As a corollary, we deduce that for all integers $a_1 \geq 1$ and $k \geq 4$ there exists $\xi \in \mathbb{R}$ such that the sequence of fractional parts $\big(\!\operatorname{frac}(\xi \alpha^n)\big)_{n \geq 0}$, where $\alpha := \big(a_1 + \sqrt{a_1^2 + 4}\,\big) / 2$, has exactly $k$ limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.