Fermat-Catalan conjecture

The Fermat–Catalan conjecture states that the equation a^m + b^n = c^k has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (a^m, b^n, c^k) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying frac 1 m + frac 1 n + frac 1 k < 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-fermat-catalan-conjecture,
  title        = {Fermat-Catalan conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/fermat-catalan-conjecture}},
  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

The Fermat–Catalan conjecture states that the equation $a^{m} + b^{n} = c^{k}$ has only finitely many solutions $(a, b, c, m, n, k)$ with distinct triplets of values $(a^{m}, b^{n}, c^{k})$ where $a, b, c$ are positive coprime integers and $m, n, k$ are positive integers satisfying $\frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.FermatCatalanConjecture.

theorem fermat_catalan : fermatCatalanConjecture

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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

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.