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

Abstract

For every monic polynomial $f\in \mathbb{Z}[X]$ with $\mathrm{deg}(f)\ge 1$, 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\{ρ(\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 $a_1\ge 1$ and $k\ge 4$ there exists $\xi \in \mathbb{R}$ such that the sequence of fractional parts $(\mathrm{frac}(\xi {\alpha }^n){)}_{n\ge 0}$, where $\alpha :=(a_1+\sqrt{a_1^2+4})/2$, has exactly $k$ limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.