====== Hall's conjecture ======

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 x<sup>3</sup> − y<sup>2</sup> 6 ≠ 0, then |x<sup>3</sup> − y<sup>2</sup>| > c(ε)x<sup>1/2−ε  </sup>([[https://web.math.pmf.unizg.hr/~duje/pdf/hall3.pdf|ref.]])

It remains an open question.

Technical details [[https://arxiv.org/pdf/1005.3448|here]] (arXiv.org)