Contributions to the theory of Diophantine equations II. The Diophantine equation y 2 = x 3 + k

Abstract

This paper is a sequel to Part I (Baker 1968) in which an effective algorithm was established for solving in integers x, y any Diophantine equation of the type y) = m, where ^denotes an irreducible binary form with integer coefficients and degree at least 3. Here the algorithm is utilized to obtain an explicit bound, free from unknown constants, for the size of all the solutions of the equation. As a consequence of the cubic case of the result, it is proved that, for any integer 4= 9, all integers x, y satisfying the equation of the title have absolute values at most exp { (10101 A:|)10 }.