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 positive $\epsilon$ 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 this [link][1]. 

See also Noam Elkies' page,  http://www.math.harvard.edu/~elkies/hall.html 


  [1]: http://en.wikipedia.org/wiki/Marshall_Hall's_conjecture