Diophantine equations with linear recurrences An overview of some recent progress

Umberto Zannier

doi:10.5802/jtnb.499

Umberto Zannier ¹

¹ Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, ITALY

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 423-435.

Abstract (VO)
Résumé

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. " $d$ -th root problem"), which in short asks whether the integrality of the values of the quotient (resp. $d$ -th root) of two (resp. one) linear recurrences implies that this quotient (resp. $d$ -th root) is itself a recurrence. We shall also relate such questions with certain natural diophantine equations, which in turn come from the simplest unknown cases of Vojta’s conjecture for integral points on algebraic varieties.

Nous discutons quelques problèmes habituels concernant l’arithmétique des suites récurrentes linéaires. Après avoir brièvement rappelé les questions et résultats anciens concernant les zéros, nous nous focalisons sur les progrès récents pour le “problème quotient” (resp. “problème de la racine $d$ -ième”), qui, pour faire court, demande si l’intégralité des valeurs du quotient (resp. racine $d$ -ième) de deux (resp. d’une) suites récurrentes linéaires entraine que ce quotient (resp. racine $d$ -ième) est lui-même une suite récurrente linéaire. Nous relions également ces questions à certaines équations diophantiennes naturelles, qui par ailleurs proviennent du cas non résolu le plus simple de la conjecture de Vojta sur les points entiers des variétés algébriques.

MR Zbl

DOI : 10.5802/jtnb.499

Affiliations des auteurs :

Umberto Zannier ¹

¹ Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, ITALY

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Umberto Zannier. Diophantine equations with linear recurrences An overview of some recent progress. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 423-435. doi : 10.5802/jtnb.499. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.499/

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