The Goormaghtigh conjecture

The only Goormaghtigh numbers are 31 and 8191.

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-goormaghtigh,
  title        = {The Goormaghtigh conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/goormaghtigh}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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

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.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Goormaghtigh.

theorem goormaghtigh_conjecture (N : ℕ) (hN : IsGoormaghtighNumber N) :
    N = 31 ∨ N = 8191

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
J. Grantham, No new Goormaghtigh primes up to $1 0^{700}$

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.