The Thue equation

Abstract

At last, we pass to the analysis of diophantine equations, starting with the central problem of the representation of numbers by binary forms. We return as in Chapter I, to the connection between the magnitude of solutions of Thue's equation and rational approximation of algebraic numbers: but now our approach is the opposite of Thue's: we obtain bounds for the approximation as a corollary to bounds for the solutions. We arrive at an effective improvement of Liouville's inequality and its generalisations; and we will see how fundamental parameters of the equation, in particular the height of the form and of the number represented by the form, influence the magnitude of the solutions.