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

Differences between perfect powers : the Lebesgue-Nagell Equation