Chapter VII - Thue equations

Summary

In this chapter we examine the method of Tzanakis and de Weger for Thue equations. In later chapters the method will be generalized to other types of diophantine equations such as Thue–Mahler equations and discriminant form equations. What is interesting about the following algorithm is that it was the first practical generic method to solve a wide class of diophantine equations. Much theoretical work had been done on bounding the solutions to such equations, but until the advent of Tzanakis and de Weger's algorithm it was not possible to produce a general algorithm, which could be applied in practice to equations of interest. There were, however, a variety of ad hoc techniques such as those to be found in [138, Chapter 23], [49], [188], [200], [199] and [190]. These either used special properties of the equations or were Skolem's method in disguise.

Some authors, see [186] and [149], had previously used the LLL–algorithm to solve specific problems or special cases. However, Tzanakis and de Weger were the first to describe the method in complete generality. There is a good survey of the start of the art up to around 1988 by A. Pethő in [144].