Differences between perfect powers : the Lebesgue-Nagell Equation

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 $n\ge 3$, with the property that $y^n>x^2$ and $x^2-y^n$ has no prime divisor exceeding $11$.

Cited by in corpus (1)

• Asymptotic Fermat's Last Theorem for a family of equations of signature $(2,2n,n)$