Hall's conjecture — open problem in number theory

Hall's conjecture

Original Hall's conjecture with exponent 1/2.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-hall,
  title        = {Hall's conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/hall}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

hall_conjecture. Original Hall's conjecture with exponent $1/2$ .

weak_hall_conjecture. Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - ε$ .

There exists a positive number $C$ such that for any integer $x, y$ with $y^{2} \neq = x^{3}$ , $∣ y^{2} - x^{3} ∣ > C ∣ x ∣$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Hall (2 statements).

theorem hall_conjecture : HallConjectureExp 2⁻¹

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

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
L. Danilov, The Diophantine equation $x^{3} - y^{2} = k$ and Hall's conjecture, Mathematical notes of the Academy of Sciences of the USSR 32 (1982): 617-618

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.