pillai_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.

Pillai's Conjecture: For every positive integer $c$, the equation $a^x - b^y = c$ has only finitely many solutions in positive integers $a, b, x, y$ with $x, y \geq 2$.

For $c = 1$: this is Catalan's conjecture, proved by Mihailescu (2002): the only solution to $a^x - b^y = 1$ with $x, y \geq 2$ is $3^2 - 2^3 = 1$.

Proposed by S. S. Pillai in 1936. Follows from the abc conjecture. It is the generalization of Catalan's conjecture to arbitrary differences $c$.

Why this node was retired

The posted statement is

theorem pillai_conjecture (c : ℕ) (hc : 0 < c) :
    Set.Finite {t : ℕ × ℕ × ℕ × ℕ |
      let (a, x, b, y) := t
      1 < x ∧ 1 < y ∧ 0 < a ∧ 0 < b ∧
      a ^ x - b ^ y = c} := by
  sorry

The set counts exponent-labeled tuples while allowing base1, creating infinitely many labels for one power value.

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

Proposed corrected statement

Count pairs of distinct perfect-power values with fixed positive difference, or require both bases>1 when counting tuples and state the corresponding finite-representation convention.

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.