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A repulsion motif in Diophantine equations.

Everest, G. and Ward, T. (2011) 'A repulsion motif in Diophantine equations.', American mathematical monthly., 118 (7). pp. 584-598.

Abstract

Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a non-singular plane cubic equation has only finitely many integral solutions. Examples show that simple equations can have inordinately large integral solutions in comparison to the size of their coefficients. A conjecture of Hall attempts to ameliorate this by bounding the size of integral solutions simply in terms of the coefficients of the defining equation. It turns out that a similar phenomenon seems, conjecturally, to be at work for solutions which are close to being integral in another sense. We describe these conjectures as an illustration of an underlying motif - repulsion - in the theory of Diophantine equations.

Item Type:Article
Full text:PDF - Accepted Version (348Kb)
Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.4169/amer.math.monthly.118.07.584
Record Created:12 Oct 2012 09:50
Last Modified:18 Oct 2012 16:30

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