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arXiv · 1005.0315

The Repulsion Motif in Diophantine Equations

Graham Everest·Thomas Ward
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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.

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Graham Everest, Thomas Ward. 2010-05-03. The Repulsion Motif in Diophantine Equations. https://doi.org/10.4169/amer.math.monthly.118.07.584

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