A repulsion motif in Diophantine equations

Everest, G.; Ward, T.

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.

Citation

Everest, G., & Ward, T. (2011). A repulsion motif in Diophantine equations. The American Mathematical Monthly, 118(7), 584-598. https://doi.org/10.4169/amer.math.monthly.118.07.584

• Journal Article Type: Article
• Publication Date: Aug 1, 2011
• Deposit Date: Oct 12, 2012
• Publicly Available Date: Oct 18, 2012
• Journal: American Mathematical Monthly
• Print ISSN: 0002-9890
• Electronic ISSN: 1930-0972
• Publisher: Mathematical Association of America (MAA)
• Peer Reviewed: Peer Reviewed
• Volume: 118
• Issue: 7
• Pages: 584-598
• DOI: https://doi.org/10.4169/amer.math.monthly.118.07.584
• Public URL: https://durham-repository.worktribe.com/output/1472347