Hall's (weak) conjecture is the statement that for all $\varepsilon > 0$ there exists a positive number $c(\varepsilon) > 0$ such that for all $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$, that

$$\displaystyle \left \lvert y^2 - x^3 \right \rvert \geq c(\varepsilon) |x|^{\frac{1}{2} - \varepsilon}.$$

My question concerns a related question. What is the smallest $\theta \geq 0$ for which it is known that there exist infinitely many $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$ satisfying

$$\displaystyle |y^2 - x^3| \ll |x|^{\frac{1}{2} + \theta}?$$

The choice $\theta = 1/2$ can be achieved as follows: write $x^3 = (x-1)^2(x+2) + 3x - 2$, and choose $x = t^2 - 2$ and $y = (t^2 - 3)t$. Then

$$y^2 - x^3 = (t^2 - 3)^2t^2 - (t^2 - 3)^2 t^2 + 3(t^2 - 2) - 2 = 3t^2 - 8,$$

which is manifestly $O(t^2) = O(|x|)$. Here we can freely choose $t \in \mathbb{Z}$, so this gives infinitely many integers satisfying the required inequality.

Can one do better than $\theta = 1/2$?


1 Answer 1


The best $\theta$ is $0$. It is known that there are infinitely many solutions of 0 < $|x^3 - y^2| \ll x^{1/2}$, parametrized by certain "Pell equations"; indeed one such family attains $|x^3 - y^2| \sim C x^{1/2}$ with $C = 5^{-5/2} 54 \approx 0.966$. See [D]. The underlying polynomial identity $$ (t^2 + 10t + 5)^3 - (t^2 + 22t + 125)(t^2 + 4t − 1)^2 = 1728t $$ is connected with the degree-6 cover ${\rm X}_0(5) \to {\rm X}(1)$ of modular curves [E].


[D] Danilov, L.V.: The Diophantine equation $x^3 - y^2 = k$ and Hall's conjecture, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.

[E] Elkies, N.D.: Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction, Lecture Notes in Computer Science 1838 (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63 (arXiv: math.NT/0005139).


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