The Goormaghtigh conjecture

formal-conjectures · wikipedia · ams-11

A repunit is a number whose digits in some base are all $1$. Here a nontrivial representation has
at least three digits. The Goormaghtigh conjecture says that $31$ and $8191$ are the only numbers
having nontrivial repunit representations in two different bases.

The only Goormaghtigh numbers are $31$ and $8191$.

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/Goormaghtigh_conjecture
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://math.colgate.edu/~integers/y98/y98.pdf
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/Goormaghtigh.lean
Local target: Corpus.WikipediaGoormaghtigh.goormaghtigh_conjecture
Source SHA-256: a711749ec6a8fdaf15e338175b890a5ebddc2fa2243c88314ea5fedfdc4e8109
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