A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

The two fundamental finiteness theorems in the arithmetic theory of elliptic curves are the Mordell-Weil theorem, which says that the group of rational points is finitely generated, and Siegel's theorem, which asserts that the set of integral points (on any affine subset) is finite. Serge Lang ([4], p. 140) has conjectured a quantitative relationship between these two results, namely that for a given number field, the number of integral points on a quasi-minimal model of an elliptic curve should be bounded solely in terms of the rank of the group of rational points. In this note we will prove Lang's conjecture for the class of elliptic curves having integral j-invariant. More precisely, we will prove the following result. (See corollary 7. 2.)