On the number of residues of certain second-order linear recurrences
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911758411628544 |
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| author | Accossato, Federico Sanna, Carlo |
| author_facet | Accossato, Federico Sanna, Carlo |
| contents | 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\{ρ(\mathbf{x}; m) : \mathbf{x} \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big\} , \end{equation*} where $ρ(\mathbf{x}; m)$ 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 $ξ\in \mathbb{R}$ such that the sequence of fractional parts $\big(\!\operatorname{frac}(ξα^n)\big)_{n \geq 0}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of residues of certain second-order linear recurrences Accossato, Federico Sanna, Carlo Number Theory 11B37 (Primary) 11B39, 11B50, 11K16 (Secondary) 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\{ρ(\mathbf{x}; m) : \mathbf{x} \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big\} , \end{equation*} where $ρ(\mathbf{x}; m)$ 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 $ξ\in \mathbb{R}$ such that the sequence of fractional parts $\big(\!\operatorname{frac}(ξα^n)\big)_{n \geq 0}$, 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. |
| title | On the number of residues of certain second-order linear recurrences |
| topic | Number Theory 11B37 (Primary) 11B39, 11B50, 11K16 (Secondary) |
| url | https://arxiv.org/abs/2401.07661 |