Hall conjectured the existence of a positive constant $C$ such that if $y^2\ne x^3$ then $$|y^2-x^3|\gt C\sqrt{|x|}$$ This has not been disproved, but is considered unlikely to be true. Nowadays one often calls Hall's Conjecture the weaker statement that for any positice $\epsilon$ there is a constant $c(\epsilon)$ such that if $y^2\ne x^3$ then $$|y^2-x^3|\gt C(\epsilon)x^{{1\over2}-\epsilon}$$ See http://en.wikipedia.org/wiki/Marshall_Hall's_conjecture.
See also Noam Elkies' page, http://www.math.harvard.edu/~elkies/hall.html