Integer points on elliptic curves

We show that the number of integer points on an elliptic curve y² = f(x) with X₀ < x ≤ X₀ + X is 蠐 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 x³ + y² = n is 蠐 n^1/6.