On the number of residues of certain second-order linear recurrences
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 \rho (x; m) : x \in \mathcal{L}(f), \, m \in \math...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.647/ |
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| Summary: | 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 \rho (x; m) : x \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big \rbrace ,
\end{equation*}
where $\rho (x; m)$ 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 $\xi \in \mathbb{R}$ such that the sequence of fractional parts $\bigl (\mathrm{frac}(\xi \alpha ^n)\big )_{n \ge 0}$, 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. |
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| ISSN: | 1778-3569 |