Catalan's conjecture and related Diophantine equations

For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation ax^n - by^m = c where (m, n) ≠ (2, 2) and x, y > 1.

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-catalan,
  title        = {Catalan's conjecture and related Diophantine equations},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/catalan}},
  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

pillais_conjecture. For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation $a x^{n} - b y^{m} = c$ where $(m, n) \neq = (2, 2)$ and $x, y > 1$ .

lebesgue_nagell. Lebesgue-Nagell Equation Conjecture

For any odd prime $p$ , the only integer solutions $(x, y)$ to the equation $x^{2} - 2 = y^{p}$ are $(x, y) = (\pm 1, - 1)$ .

Reference: Ethan Katz and Kyle Pratt, "On the Lebesgue-Nagell equation $x^{2} - 2 = y^{p}$ ", arXiv:2507.12397

Formal statement (Lean 4)

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

theorem pillais_conjecture (a b c : ℕ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
    { (x, y, m, n) : (ℕ × ℕ × ℕ × ℕ) |
      1 < x ∧ 1 < y ∧ 1 < m ∧ 1 < n ∧ (m, n) ≠ (2, 2) ∧
      a * x^n - b * y^m = c }.Finite

theorem lebesgue_nagell (p : ℕ) (hp : p.Prime) (hodd : Odd p) (x y : ℤ) :
    x ^ 2 - 2 = y ^ p ↔ (x = 1 ∨ x = -1) ∧ y = -1

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 - Catalan's conjecture
arXiv:2507.12397 (Lebesgue-Nagell equation)

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.