Integer points on elliptic curves

Abstract

We show that the number of integer points on an elliptic curve $y^2=f(x)$ with $X_0\lt x\le X_0+X$ is $\ll X^{1/2}$ where the implicit constant depends at most on the degree of $f(x)$. This improves on various bounds of Cohen \cite{SDC}, Bombieri and Pila \cite{BP} and of Pila \cite{PJ}, and others. In particular it follows that the number of positive integral solutions to $x^3+y^2=n$ is $\ll n^{1/6}$.