Catalan's conjecture and related Diophantine equations

formal-conjectures · wikipedia · ams-11

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 $(m, n) \neq (2, 2)$ and $x, y > 1$.

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/Catalan%27s_conjecture
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/2507.12397
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/Catalan.lean
Local target: Corpus.WikipediaCatalan.pillais_conjecture
Source SHA-256: ea5b13bc7c5224b2c5d1645d6e904acbe432e573e05294f25c0af748b94e4caa
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.

Public JSON record