G. Everest
A repulsion motif in Diophantine equations
Everest, G.; Ward, T.
Authors
T. Ward
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 |
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