formal-conjectures · wikipedia · ams-11
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
$\frac 1 m + \frac 1 n + \frac 1 k < 1$.
Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).
Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.
Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Fermat-Catalan_conjecture
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/FermatCatalanConjecture.lean
Local target: Corpus.WikipediaFermatCatalanConjecture.fermat_catalan
Source SHA-256: f4b414f5b1c7719807d84570ba8a65859f917ce03ab019adee50f92ad6bbac22
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.
Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.
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
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