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

Find positive integers x, y with 0 < |y² − x³| < √x/100, beating the record ratio of about 46.6 in Hall's conjecture, unbroken since 1998.

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State the problem, from FrontierMath: Open Problems

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	1	+# Hall's conjecture: a square extremely close to a cube
	2	+
	3	+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.
	4	+
	5	+## Problem
	6	+
	7	+For positive integers $x, y$ with $y^2 \ne x^3$, define the Hall ratio
	8	+$$r(x, y) = \frac{\sqrt{x}}{\left\| y^2 - x^3 \right\|}.$$
	9	+Hall's original conjecture asserts that $r(x, y)$ is bounded by an absolute constant.
	10	+
	11	+Task. Find positive integers $x, y$ with $y^2 \ne x^3$ and $r(x, y) > 100$, that is, $x > \left(100\, \left\| y^2 - x^3 \right\|\right)^2$.
	12	+
	13	+## Background
	14	+
	15	+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].
	16	+
	17	+## What is known
	18	+
	19	+- 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.
	20	+- 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].
	21	+- formal-conjectures states the original conjecture in Lean as `hall_conjecture` in [Hall.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/Hall.lean), together with Elkies' example.
	22	+
	23	+## What would count as progress
	24	+
	25	+- 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.
	26	+- 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.
	27	+
	28	+## References
	29	+
	30	+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](https://doi.org/10.1007/10722028_2) ([arXiv:math/0005139](https://arxiv.org/abs/math/0005139))
	31	+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](https://people.math.harvard.edu/~elkies/hall.html)
	32	+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](https://doi.org/10.1090/S0025-5718-09-02240-6)
	33	+
	34	+## Source
	35	+
	36	+Imported from [FrontierMath: Open Problems, hall-conjecture](https://epoch.ai/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.