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Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation

Published online by Cambridge University Press:  13 January 2006

Yann Bugeaud ,
Maurice Mignotte  and
Samir Siksek
Yann Bugeaud
Affiliation:
Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg cedex, Francebugeaud@math.u-strasbg.fr
Maurice Mignotte
Affiliation:
Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg cedex, Francemignotte@math.u-strasbg.fr
Samir Siksek
Affiliation:
Department of Mathematics, University of Warwick, Coventry, CV4 7AL, UKsiksek@maths.warwick.ac.uk
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Abstract

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This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue–Nagell equation x2 + D = yn, x, y integers, $n\geq 3$, for D in the range $1 \leq D \leq 100$.

Keywords

Diophantine equationsRamanujan–NagellFrey curveslevel-loweringlinear forms in logarithmsThue equations11D6111J86 (primary)11D5911Y50 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 1 , January 2006 , pp. 31 - 62
Copyright
Foundation Compositio Mathematica 2006