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

By Yann Bugeaud, Maurice Mignotte, Samir Siksek. Compositio Mathematica, 2004

181 citations recorded by the source corpus.

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 $1 \leq D \leq 100$.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/638cb57fb19d92ffe6bf41de1266418a04efead8

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Citations in the corpus

There is no Diophantine quintuple (2016, background): “Using deep results on lower bounds for linear forms in three logarithms due to Mignotte [37] (see also [5, 6]) we can further improve the bound and finally arrive at the upper bounds d < 1.”
The Modular Approach to Diophantine Equations (2007, methodology): “A combination of Proposition 9.1, the method of Kraus and classical techniques for Diophantine equations recently lead to the complete solutions of equations x2 + D = yn for n 3 and 1 D 100 (see [ 9 ],”
Fibonacci numbers at most one away from a perfect power (2008, background): “The problem of determining all perfect powers in the Fibonacci sequence was a famous open problem for over 40 years, and has been resolved only recently [9].”
Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers (2004, methodology): “A sequel combining the classical and modular approaches for Diophantine equations of the form x2 + D = yp has just been completed [13].”