Differences between perfect powers : the Lebesgue-Nagell Equation

TL;DR

This paper introduces new techniques combining logarithmic bounds, modularity, and Diophantine approximation to analyze solutions of the equation x^2 + D = y^n, explicitly characterizing solutions with certain prime divisor constraints.

Contribution

It develops novel methods to solve specific exponential Diophantine equations, explicitly determining solutions under prime divisor restrictions.

Findings

Explicitly determined all coprime solutions with y^n > x^2 and prime divisors of x^2 - y^n ≤ 11.

Established bounds for solutions using advanced number-theoretic techniques.

Connected modularity of elliptic curves to Diophantine equation solutions.

Abstract

We develop a variety of new techniques to treat Diophantine equations of the shape $x^2+D =y^n$, based upon bounds for linear forms in $p$-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers $x$ and $y$, and $n \geq 3$, with the property that $y^n > x^2$ and $x^2-y^n$ has no prime divisor exceeding $11$.