Hall's conjecture: a square extremely close to a cube

How close can a perfect square be to a perfect cube without being equal to it? Marshall Hall Jr. asked in 1971 how small $|y^2 - x^3|$ can be, and conjectured that it is always at least a constant times $\sqrt{x}$. The best example known, found by Noam Elkies, comes within $\sqrt{x}/46.6$, and the task is to beat that ratio by a wide margin.

Problem

For positive integers $x, y$ with $y^2 \ne x^3$, define the Hall ratio

$$r(x, y) = \frac{\sqrt{x}}{\left| y^2 - x^3 \right|}.$$

Hall's original conjecture asserts that $r(x, y)$ is bounded by an absolute constant.

Task. Find positive integers $x, y$ with $y^2 \ne x^3$ and $r(x, y) > 100$, that is, $x > \left(100\, \left| y^2 - x^3 \right|\right)^2$.

Background

Hall raised the question in his 1971 paper "The Diophantine equation $x^3 - y^2 = k$". The weaker form, $\left| x^3 - y^2 \right| > x^{1/2 - o(1)}$, is now recognised as a special case of the abc conjecture. Elkies calls the original strong form almost certainly false, but refuting it needs examples with ever larger ratios [2]. Elkies found an algorithm, based on lattice reduction, that finds all solutions of $\left| x^3 - y^2 \right| \ll x^{1/2}$ with $x < N$ in time $O(N^{1/2 + o(1)})$. Running it up to $N = 10^{18}$ produced ten new examples with $\left| x^3 - y^2 \right| < \sqrt{x}$, including one that broke the previous record ratio by a factor of nearly 10 [1, 2].

What is known

• The record is $r \approx 46.6$, from $x = 5853886516781223$, $y = 447884928428402042307918$, $x^3 - y^2 = 1641843$, found by Elkies [2]; Epoch dates it to 1998 and notes that it has not been improved since.
• Elkies's page lists every solution of $0 < \left| x^3 - y^2 \right| < \sqrt{x}$ with $x < 10^{18}$; the previous record ratio was $4.87$ [2]. Later work developed new search algorithms [3].
• formal-conjectures states the original conjecture in Lean as hall_conjecture in Hall.lean, together with Elkies' example.

What would count as progress

• A pair $(x, y)$ with $x > \left(100 \left| y^2 - x^3 \right|\right)^2$, of any size, checked exactly in integer arithmetic; this is the test Epoch's verifier applies.
• Any pair with ratio above 46.6 would already be a new record; a method that produces many good pairs would shed light on how $x^3 - y^2$ behaves.

References

1. N. D. Elkies, Rational points near curves and small nonzero $\left| x^3 - y^2 \right|$ via lattice reduction, ANTS-IV, Lecture Notes in Comput. Sci. 1838 (2000), 33–63. doi:10.1007/10722028_2 (arXiv:math/0005139)
2. N. D. Elkies, List of integers $x, y$ with $x < 10^{18}$ and $0 < \left| x^3 - y^2 \right| < x^{1/2}$ (web page). people.math.harvard.edu/~elkies/hall.html
3. I. Jiménez Calvo, J. Herranz and G. Sáez, A new algorithm to search for small nonzero $\left| x^3 - y^2 \right|$ values, Math. Comp. 78 (2009), 2435–2444. doi:10.1090/S0025-5718-09-02240-6

Source

Imported from FrontierMath: Open Problems, hall-conjecture, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.