Almost powers in the Lucas sequence

Bugeaud, Yann; Luca, Florian; Mignotte, Maurice; Siksek, Samir

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 555-600.

La liste complète des puissances pures qui apparaissent dans les suites de Fibonacci ${\left(F_n\right)}_{n\ge 0}$ et de Lucas ${\left(L_n\right)}_{n\ge 0}$ ne fut déterminée que tout récemment [10]. Les démonstrations combinent des techniques modulaires, issues de la preuve de Wiles du dernier théorème de Fermat, avec des méthodes classiques d’approximation diophantienne, dont la théorie de Baker. Dans le présent article, nous résolvons les équations diophantiennes $L_n=q^a{y}^p$, avec $a>0$ et $p\ge 2$, pour tous les nombres premiers $q<1087$, et en fait pour tous les nombres premiers $q<{10}^6$ à l’exception de $13$ d’entre eux. La stratégie suivie dans [10] s’avère inopérante en raison de la taille des bornes numériques données par les méthodes classiques et de la complexité des équations de Thue qui apparaissent dans notre étude. La nouveauté mise en avant dans le présent article est l’utilisation simultanée de deux courbes de Frey afin d’aboutir à des équations de Thue plus simples, et donc à de meilleures bornes numériques, qui contredisent les minorations que donne le crible modulaire.

The famous problem of determining all perfect powers in the Fibonacci sequence ${\left(F_n\right)}_{n\ge 0}$ and in the Lucas sequence ${\left(L_n\right)}_{n\ge 0}$ 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 $a>0$ and $p\ge 2$, 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.

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