Diophantine equations with linear recurrences An overview of some recent progress

Umberto Zannier[1]

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

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 423-435
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Zannier, Umberto. "Diophantine equations with linear recurrences An overview of some recent progress." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 423-435. <http://eudml.org/doc/249473>.

@article{Zannier2005,
abstract = {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.},
affiliation = {Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, ITALY},
author = {Zannier, Umberto},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {423-435},
publisher = {Université Bordeaux 1},
title = {Diophantine equations with linear recurrences An overview of some recent progress},
url = {http://eudml.org/doc/249473},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Zannier, Umberto
TI - Diophantine equations with linear recurrences An overview of some recent progress
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 423
EP - 435
AB - 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.
LA - eng
UR - http://eudml.org/doc/249473
ER -

References

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