Almost powers in the Lucas sequence

Yann Bugeaud[1]; Florian Luca[2]; Maurice Mignotte[1]; Samir Siksek[3]

  • [1] Université Louis Pasteur U. F. R. de mathématiques 7, rue René Descartes 67084 Strasbourg Cedex, France
  • [2] Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
  • [3] Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 3, page 555-600
  • ISSN: 1246-7405

Abstract

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The famous problem of determining all perfect powers in the Fibonacci sequence ( F n ) n ≥ 0 and in the Lucas sequence ( L n ) n ≥ 0 has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations L n = q a y p , with a > 0 and p ≥ 2 , for all primes q < 1087 and indeed for all but 13 primes q < 10 6 . Here the strategy of [10] is not sufficient due to the sizes of the bounds and complicated nature of the Thue equations involved. The novelty in the present paper is the use of the double-Frey approach to simplify the Thue equations and to cope with the large bounds obtained from Baker’s theory.

How to cite

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Bugeaud, Yann, et al. "Almost powers in the Lucas sequence." Journal de Théorie des Nombres de Bordeaux 20.3 (2008): 555-600. <http://eudml.org/doc/10852>.

@article{Bugeaud2008,
abstract = {The famous problem of determining all perfect powers in the Fibonacci sequence $(F_n)_\{n\ge 0\}$ and in the Lucas sequence $(L_n)_\{n\ge 0\}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations $L_n=q^a y^p$, with $a&gt;0$ and $p\ge 2$, for all primes $q&lt;1087$ and indeed for all but $13$ primes $q &lt; 10^6$. Here the strategy of [10] is not sufficient due to the sizes of the bounds and complicated nature of the Thue equations involved. The novelty in the present paper is the use of the double-Frey approach to simplify the Thue equations and to cope with the large bounds obtained from Baker’s theory.},
affiliation = {Université Louis Pasteur U. F. R. de mathématiques 7, rue René Descartes 67084 Strasbourg Cedex, France; Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México; Université Louis Pasteur U. F. R. de mathématiques 7, rue René Descartes 67084 Strasbourg Cedex, France; Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom},
author = {Bugeaud, Yann, Luca, Florian, Mignotte, Maurice, Siksek, Samir},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Lucas sequence; modular method; double Frey method; linear form in two or three logarithms; Thue equation},
language = {eng},
number = {3},
pages = {555-600},
publisher = {Université Bordeaux 1},
title = {Almost powers in the Lucas sequence},
url = {http://eudml.org/doc/10852},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Bugeaud, Yann
AU - Luca, Florian
AU - Mignotte, Maurice
AU - Siksek, Samir
TI - Almost powers in the Lucas sequence
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 3
SP - 555
EP - 600
AB - The famous problem of determining all perfect powers in the Fibonacci sequence $(F_n)_{n\ge 0}$ and in the Lucas sequence $(L_n)_{n\ge 0}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations $L_n=q^a y^p$, with $a&gt;0$ and $p\ge 2$, for all primes $q&lt;1087$ and indeed for all but $13$ primes $q &lt; 10^6$. Here the strategy of [10] is not sufficient due to the sizes of the bounds and complicated nature of the Thue equations involved. The novelty in the present paper is the use of the double-Frey approach to simplify the Thue equations and to cope with the large bounds obtained from Baker’s theory.
LA - eng
KW - Lucas sequence; modular method; double Frey method; linear form in two or three logarithms; Thue equation
UR - http://eudml.org/doc/10852
ER -

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