Differences between perfect powers : the Lebesgue-Nagell Equation

Abstract: We develop a variety of new techniques to treat Diophantine equations of the shape $x^2+D =y<sup>n$, 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 $yⁿ &gt; x<sup>2$ and $x<sup>2-y<sup>n$ has no prime divisor exceeding $11$.