In previous chapters we have seen how to solve the problem of finding all integral points on an elliptic curve. The methods used either a reduction to a finite set of Thue equations or reduction to a finite set of S-unit equations. These methods had numerous drawbacks in that they involved using expensive computations in number fields and they ignored much of the beauty of elliptic curves.

In this chapter we present a much better method which uses a lot of the underlying structure of an elliptic curve. The new method is based on the method of elliptic logarithms. The idea behind this method can be found in a paper by Lang from 1964 [110]. It is also explained in [111] or [172], and an outline of the method was also given in [219]. However, it was not until David [44] gave an explicit transcendence result for elliptic logarithms that it became a general method. This method is now the standard one, which is apparent from looking at the relevant, literature [191], [77], [179], [192], [189], [201] and [185].

There is one drawback with the new method in that we need to be able to compute the Mordell–Weil group. In other words to find all integral points we shall need an explicit description of the set of all rational points on the curve.