Open Access
2014 Integer points on elliptic curves
R.C. Vaughan
Rocky Mountain J. Math. 44(4): 1377-1382 (2014). DOI: 10.1216/RMJ-2014-44-4-1377


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 $\ll n^{1/6}$.


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R.C. Vaughan. "Integer points on elliptic curves." Rocky Mountain J. Math. 44 (4) 1377 - 1382, 2014.


Published: 2014
First available in Project Euclid: 31 October 2014

zbMATH: 1311.11048
MathSciNet: MR3274354
Digital Object Identifier: 10.1216/RMJ-2014-44-4-1377

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.44 • No. 4 • 2014
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