goormaghtigh_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Goormaghtigh conjecture (1917): The only solutions to (x^m-1)/(x-1) = (y^n-1)/(y-1) with x ≠ y and m,n ≥ 2 are (x,m,y,n) = (2,5,5,3) giving 31, and (2,13,90,3) giving 8191. Only two known solutions; proving no others exist is open.

Why this node was retired

The posted statement is

import Mathlib

theorem goormaghtigh_conjecture :
    ∀ (x y m n : ℕ), 2 ≤ x → 2 ≤ y → x ≠ y → 2 ≤ m → 2 ≤ n →
      (x^m - 1) / (x - 1) = (y^n - 1) / (y - 1) →
      (x = 2 ∧ y = 5 ∧ m = 5 ∧ n = 3) ∨ (x = 2 ∧ y = 90 ∧ m = 13 ∧ n = 3) := by
  sorry

The exceptional solutions use one orientation but the hypotheses do not impose that ordering.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Impose the matching convention x<y, or include the swapped exceptional tuples as additional alternatives while preserving the original lower bounds.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.