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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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.647/ |
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| author | Accossato, Federico Sanna, Carlo |
| author_facet | Accossato, Federico Sanna, Carlo |
| author_sort | Accossato, Federico |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-c1a7e977605f40109447dc2603ee86f2 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-c1a7e977605f40109447dc2603ee86f22025-02-07T11:23:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G111365137710.5802/crmath.64710.5802/crmath.647On the number of residues of certain second-order linear recurrencesAccossato, Federico0https://orcid.org/0009-0004-9861-0153Sanna, Carlo1https://orcid.org/0000-0002-2111-7596Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, ItalyDepartment of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, ItalyFor 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.647/Fractional parts of powersLehmer sequenceslinear recurrencesPisot numbersprimitive divisorsresidues |
| spellingShingle | Accossato, Federico Sanna, Carlo On the number of residues of certain second-order linear recurrences Comptes Rendus. Mathématique Fractional parts of powers Lehmer sequences linear recurrences Pisot numbers primitive divisors residues |
| title | On the number of residues of certain second-order linear recurrences |
| title_full | On the number of residues of certain second-order linear recurrences |
| title_fullStr | On the number of residues of certain second-order linear recurrences |
| title_full_unstemmed | On the number of residues of certain second-order linear recurrences |
| title_short | On the number of residues of certain second-order linear recurrences |
| title_sort | on the number of residues of certain second order linear recurrences |
| topic | Fractional parts of powers Lehmer sequences linear recurrences Pisot numbers primitive divisors residues |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.647/ |
| work_keys_str_mv | AT accossatofederico onthenumberofresiduesofcertainsecondorderlinearrecurrences AT sannacarlo onthenumberofresiduesofcertainsecondorderlinearrecurrences |