Majorations Effectives Pour L’ Équation de Fermat Généralisée

Kraus, Alain

doi:10.4153/cjm-1997-056-2

Cited by 61 publications

(133 citation statements)

References 8 publications

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“…In the above proof, we used the following result of Kraus [19,Proposition 8.1] on congruences of modular forms.…”

Section: The Newforms Have Fourier Expansions Around the Cusp At Infimentioning

confidence: 99%

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Almost powers in the Lucas sequence

Bugeaud¹,

Luca²,

Mignotte³

et al. 2008

J. Théor. Nombres Bordeaux

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A Henri Cohen,à l'occasion de son soixantième anniversaire Résumé. La liste complète des puissances pures qui apparaissent dans les suites de Fibonacci (F n ) n≥0 et de Lucas (L n ) n≥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 leséquations diophantiennes L n = q a y p , avec a > 0 et p ≥ 2, pour tous les nombres premiers q < 1087, et en fait pour tous les nombres premiers q < 10 6à 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 deséquations de Thue qui apparaissent dans notreétude. La nouveauté mise en avant dans le présent article est l'utilisation simultanée de deux courbes de Frey afin d'aboutirà deséquations de Thue plus simples, et doncà de meilleures bornes numériques, qui contredisent les minorations que donne le crible modulaire.Abstract. 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.

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“…In the above proof, we used the following result of Kraus [19,Proposition 8.1] on congruences of modular forms.…”

Section: The Newforms Have Fourier Expansions Around the Cusp At Infimentioning

confidence: 99%

“…Following the recipes in [2], we associate to solutions to equations (13) and (14) the Frey elliptic curves (19) G m : Y 2 = X 3 + 5F 2m X 2 − 5X, and (20)…”

Section: Frey Curves and Level-loweringmentioning

confidence: 99%

Almost powers in the Lucas sequence

Bugeaud¹,

Luca²,

Mignotte³

et al. 2008

J. Théor. Nombres Bordeaux

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“…Bennett [2], [3], Kraus [34], [35] and Siksek [13], [12] and their collaborators have developed and clarified the method using Frey elliptic curves over Q. Unfortunately, there is a restrictive set of exponents (p, q, r) which can be approached using the modular method over Q due to constraints coming from the classification of Frey representations [19].…”

Section: Introductionmentioning

confidence: 99%

A multi-Frey approach to Fermat equations of signature $(r,r,p)$

Billerey

Chen

Dieulefait

et al. 2019

Trans. Amer. Math. Soc.

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Abstract. In this paper, we give a resolution of the generalized Fermat equations x 5 + y 5 = 3z n and x 13 + y 13 = 3z n , for all integers n ≥ 2, and all integers n ≥ 2 which are not a power of 7, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents n.We also give a number of results for the equations x 5 + y 5 = dz n , where d = 1, 2, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at p modulo p.

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“…How to associate such an equation to a Frey curve is detailed for three important signatures (p, p, p), (p, p, 2) and (p, p, 3) respectively by Kraus [20], by Bennett and Skinner [2], and by Bennett, Vatsal and Yazdani [3]. For convenience of the reader we reproduce the recipes appearing in these papers for the Frey curves and levels.…”

Section: Recipes For Ternary Diophantine Equationsmentioning

confidence: 99%

“…In this section we take a close look at the equation (8) x p + L r y p + z p = 0, xyz = 0, p ≥ 5 is prime, studied by Serre in [29] and Kraus in [20] -the connection of this equation with Mazur will become apparent. We assume that (9) x, y, z are pairwise coprime, 0 < r < p.…”

Section: An Example Of Serre-mazur-krausmentioning

confidence: 99%

The Modular Approach to Diophantine Equations

Siksek¹

Number Theory

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Majorations Effectives Pour L’ Équation de Fermat Généralisée

Cited by 61 publications

References 8 publications

Almost powers in the Lucas sequence

Almost powers in the Lucas sequence

A multi-Frey approach to Fermat equations of signature $(r,r,p)$

The Modular Approach to Diophantine Equations

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