fermat_catalan_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.

Fermat–Catalan Conjecture: There are only finitely many solutions $(a, b, c, p, q, r)$ with $p, q, r \geq 2$, $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} < 1$, $\gcd(a,b,c) = 1$, and $a^p + b^q = c^r$.

The condition $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} < 1$ (i.e., $qr + pr + pq < pqr$) excludes trivial families. Known solutions include: $1^m + 2^3 = 3^2$ (for any $m$), $2^5 + 7^2 = 3^4$, $7^3 + 13^2 = 2^9$, $2^{14} + 17^3 = 71^2$, and a few others — only 10 families are known.

Conjectured by Darmon and Granville (1995), who also proved finitely many solutions exist for fixed exponents $(p,q,r)$. Generalizes both FLT (which is the case $p=q=r\geq 3$) and the Catalan conjecture (proved by Mihailescu in 2002).

Why this node was retired

The posted statement is

theorem fermat_catalan_conjecture :
    Set.Finite {t : ℕ × ℕ × ℕ × ℕ × ℕ × ℕ |
      let (a, b, c, p, q, r) := t
      2 ≤ p ∧ 2 ≤ q ∧ 2 ≤ r ∧
      q * r + p * r + p * q < p * q * r ∧
      0 < a ∧ 0 < b ∧ 0 < c ∧
      Nat.Coprime a b ∧ Nat.Coprime b c ∧ Nat.Coprime a c ∧
      a ^ p + b ^ q = c ^ r} := by
  sorry

The set counts sextuples, retaining a free exponent on base1, rather than triples of power values.

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

Proposed corrected statement

State finiteness of the image set of power-value triples(a^p,b^q,c^r) satisfying the original exponent and coprimality conditions, thereby identifying redundant exponents on1.

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.