A Repulsion Motif in Diophantine Equations

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 nonsingular 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. Nonetheless, a conjecture of Hall suggests a bound on the size of integral solutions 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 this conjecture as an illustration of an underlying motif—repulsion—in the theory of Diophantine equations. 1. CHALLENGING QUESTIONS. In 1657, Pierre de Fermat challenged the English mathematicians Sir Kenelm Digby and John Wallis to find all the integer solutions to the equation