Submit progress

on Weak form of Hall's conjecture

The statement

Let $\varepsilon > 0$ be a real number. A pair of real numbers $(C, e)$ with $C > 0$ is said to satisfy the Hall inequality if for all integers $x$ and $y$ with $y^2 \neq x^3$ ,

|y^2 - x^3| > C \cdot |x|^e.

The Hall conjecture with exponent $e$ holds if there exists $C > 0$ such that $(C, e)$ satisfies the Hall inequality. Prove that for every $\varepsilon > 0$ , the Hall conjecture with exponent $\frac{1}{2} - \varepsilon$ holds.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.