INTEGER POINTS ON ELLIPTIC CURVES

We show that the number of integer points on an elliptic curve y(2) = f(x) with X-0 < x <= X-0 + X is << X-1/2 where the implicit constant depends at most on the degree of f(x). This improves on various bounds of Cohen [4], Bombieri and Pila [1] and of Pila [9], and others. In particular it follows that the number of positive integral solutions to x(3) + y(2) = n is << n(1/6).