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

  • Yann Bugeaud (a1), Maurice Mignotte (a2) and Samir Siksek (a3)
Abstract

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 $1 \leq D \leq 100$.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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