Contributions to the theory of diophantine equations I. On the representation of integers by binary forms

Abstract

An effective algorithm is established for solving in integers x, y any Diophantine equation of the type/( x, y ) = m , where/ denotes an irreducible binary form with integer coefficients and degree at least 3. The magnitude, relative to m, of the bound furnished by the algorithm for the size of all the solutions of the equation is investigated, and, in consequence, there is obtained the first generally effective improvement on the well known result of Liouville (1844) concerning the accuracy with which algebraic numbers can be approximated by rationals.