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 x^2+D =y^n , based upon bounds for linear forms in p 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 x and y y , and n ≥ 3 n \geq 3 , with the property that y n > x 2 y^n > x^2 and x 2 − y n x^2-y^n has no prime divisor exceeding 11 11 .