  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.
