In 1970, American mathematician Marshall Hall Jr. conjectured that :

$${\displaystyle |y^{2}-x^{3}| >C{\sqrt {|x|}}.}$$

Or, in language form : For any ε > 0, there exists a constant c(ε) > 0 such that if x and y are positive integers satisfying x³ − y² 6 ≠ 0, then |x³ − y²| > c(ε)x^{1/2−ε} (ref.)

It remains an open question.

Technical details here (arXiv.org)