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

摘要

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 x(2) + D = y(n), x, y integers, n >= 3, for D in the range 1 <= D <= 100.