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)x1/2−ε (ref.)
It remains an open question.
Technical details here (arXiv.org)