On the number of residues of certain second-order linear recurrences - Accossato, Federico; Sanna, Carlo (Comptes Rendus. Mathématique)
For every monic polynomial $f \in \mathbb{Z}[X]$ with $\deg (f) \ge 1$, 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 \lbrace ρ (x; m) : x \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big \rbrace , \end{equation*} where $ρ (x; m)$ is the number of distinct residues of $x$ 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 $k \ge 2$ there exists $ξ \in \mathbb{R}$ such that the sequence of fractional parts $\bigl (frac(ξ α ^n)\big )_{n \ge 0}$, where $α :=\big (a_1 + \sqrt{a_1^2 + 4}\,\big ) / 2$, has exactly $k$ limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.