Computing Integral Points on Elliptic Curves

By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demjanenko [L3] states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Silverman [Si1] for elliptic curves E with integral modular invariant j over K and by Hindry and Silverman [HSi] for algebraic function fields K. On the other hand, beginning with Baker [B], bounds for the size of the coefficients of integral points on E have been found by various authors (see [L4]). The most recent bound was exhibited by W. Schmidt [Sch, Th. 2]. However, the bounds are rather large and therefore can be used only for soloving some particular equations (see [TdW], [St]) or for treating a special model of elliptic curves, namely Thue curves of degree 3 (see [GSch]).