Solving Thue equations  without the full unit group

Hanrot, Guillaume

doi:10.1090/s0025-5718-99-01124-2

Cited by 25 publications

(50 citation statements)

References 18 publications

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“…Hence, gcd(m , M l ) = 2 α . The fact that m , α satisfy (a) and (b.1-b.4) in the definition of S l is quickly deduced from conditions (17) and (18). Thus, m ∈ S l as required.…”

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

confidence: 95%

“…Suppose that r is an odd prime dividing gcd(m , M l ). Then r | m, and so from conditions (17) and (18) we deduce that r = r > 2 × 10 5 . But r | M l , so P l > 2 × 10 5 , giving a contradiction.…”

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

confidence: 96%

“…Thus, we may write v = 2v , and rewrite our equation in the form F (u, v ) = 1, where F (u, 1) is now monic. The best practical algorithms for solving Thue equations that we are aware of are those of Bilu and Hanrot [3], and of Hanrot [18]. Fortunately, these are implemented as part of the computer package PARI/GP, and using these we have been able to solve the Thue equations for 3 ≤ p ≤ 11 in under a day on a 2.4 GHz Xeon.…”

Section: Auxiliary Equationsmentioning

confidence: 99%

“…Suppose that (m, y, p) is a solution to equation (11) satisfying conditions (17), (18) and l satisfies…”

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

confidence: 99%

“…The definition of S l takes into account the information about m shown in (17) and (18). It does not take into account the information about m given by the Frey curves and the level-lowering.…”

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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“…Hence, gcd(m , M l ) = 2 α . The fact that m , α satisfy (a) and (b.1-b.4) in the definition of S l is quickly deduced from conditions (17) and (18). Thus, m ∈ S l as required.…”

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

confidence: 95%

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

confidence: 96%

Section: Auxiliary Equationsmentioning

confidence: 99%

“…Suppose that (m, y, p) is a solution to equation (11) satisfying conditions (17), (18) and l satisfies…”

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

confidence: 99%

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

confidence: 99%

See 3 more Smart Citations