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The author has been informed by Serge Lang that the conjecture that all integral solutions of y2 = x3 + ax + b are polynomially bounded in a and b (cf. [3, Section 2]), and the observation that a naive probabilistic argument would lead to the expectation that the number p = (logx)/log(max(|a|1/2,16p/3)) does not ever exceed 10 + o(l), have been made by him and by Harold Stark; Lang conjectured that all solutions, with the possible exception of a finite number of parametric families, satisfy p < 10 + o(l) (the necessity of the caveat about exceptional families followed from a related example of Stark's). For all this, see [1]. It has been shown by Paul Vojta that Lang's conjecture in this form follows from his own more general Diophantine conjecture [2]. Recently, Noam Elkies has constructed an infinite family of curves having an integral solution with p = 12 + o(l).

Addendum to: “Large integral points on elliptic curves” [Math. Comp. 48 (1987), no. 177, 425–436; MR0866125 (87k:11062)]