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

Federico Accossato; Carlo Sanna

doi:10.5802/crmath.647

Article de recherche - Théorie des nombres

On the number of residues of certain second-order linear recurrences
[Sur le nombre de résidus de certaines récurrences linéaires du second ordre]

Federico Accossato ¹ ; Carlo Sanna ¹

¹ Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1365-1377.

Résumé
Abstract

Pour chaque polynôme monique $f \in ℤ [X]$ avec $deg (f) \geq 1$ , soit $ℒ (f)$ l’ensemble de toutes les récurrences linéaires avec des valeurs dans $ℤ$ et un polynôme caractéristique $f$ , et soit

ℛ (f) : = \{ρ (x; m) : x \in ℒ (f), m \in ℤ^{+}\},

où $ρ (x; m)$ est le nombre de résidus distincts de $x$ modulo $m$ .

Dubickas et Novikas ont prouvé que $ℛ (X^{2} - X - 1) = ℤ^{+}$ . Nous généralisons ce résultat en montrant que $ℛ (X^{2} - a_{1} X - 1) = ℤ^{+}$ pour tout entier non nul $a_{1}$ . Comme corollaire, nous déduisons que pour tous les entiers $a_{1} \geq 1$ et $k \geq 2$ , il existe $ξ \in ℝ$ tel que la séquence des parties fractionnaires ${(frac (ξ α^{n}))}_{n \geq 0}$ , où $α : = (a_{1} + \sqrt{a_{1}^{2} + 4},) / 2$ , a exactement $k$ points de limite. Nos preuves sont constructives et utilisent certains résultats sur l’existence de diviseurs primitifs spéciaux de certaines séquences de Lehmer.

For every monic polynomial $f \in ℤ [X]$ with $deg (f) \geq 1$ , let $ℒ (f)$ be the set of all linear recurrences with values in $ℤ$ and characteristic polynomial $f$ , and let

ℛ (f) : = \{ρ (x; m) : x \in ℒ (f), m \in ℤ^{+}\},

where $ρ (x; m)$ is the number of distinct residues of $x$ modulo $m$ .

Dubickas and Novikas proved that $ℛ (X^{2} - X - 1) = ℤ^{+}$ . We generalize this result by showing that $ℛ (X^{2} - a_{1} X - 1) = ℤ^{+}$ for every nonzero integer $a_{1}$ . As a corollary, we deduce that for all integers $a_{1} \geq 1$ and $k \geq 2$ there exists $ξ \in ℝ$ such that the sequence of fractional parts ${(frac (ξ α^{n}))}_{n \geq 0}$ , where $α : = (a_{1} + \sqrt{a_{1}^{2} + 4}) / 2$ , has exactly $k$ limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.

Reçu le : 2024-02-23
Révisé le : 2024-04-23
Accepté le : 2024-05-03
Publié le : 2024-11-14

DOI : 10.5802/crmath.647

Classification : 11B37, 11B39, 11B50, 11K16
Keywords: Fractional parts of powers, Lehmer sequences, linear recurrences, Pisot numbers, primitive divisors, residues
Mot clés : Parties fractionnaires des puissances, suites de Lehmer, récurrences linéaires, nombres de Pisot, diviseurs primitifs, résidus

Affiliations des auteurs :

Federico Accossato ¹ ; Carlo Sanna ¹

¹ Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Federico Accossato and Carlo Sanna},
     title = {On the number of residues of certain second-order linear recurrences},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1365--1377},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.647},
     language = {en},
}

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Federico Accossato; Carlo Sanna. On the number of residues of certain second-order linear recurrences. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1365-1377. doi : 10.5802/crmath.647. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.647/

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