OP-6962Status unassessedNumber theory

Weak form of Hall's conjecture

TheoremDB P8934 · Candidate awaiting source review

Inferred from the community field; finer classification has not been reviewed.

Source candidate · not reviewedAs of 2026-09-08Source & evidence ↓

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Problem 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.

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Formal statement

Lean 4

theorem weak_hall_conjecture (ε : ℝ) (hε : ε > 0) : HallConjectureExp (2⁻¹ - ε)

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Progress & source assessment

Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.

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References

[1]TheoremDB P8934 link
[2]Source formal statement (not locally verified) link

Recorded progress

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Discussion

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