Open problemChecked July 24, 2026

Beal Conjecture

A far-reaching generalization of Fermat's Last Theorem with an active prize.

Number theory Diophantine equations Exponential equations

Mathematical statement

If positive integers satisfy $A^x+B^y=C^z$ with $x,y,z>2$ , then $A$ , $B$ , and $C$ share a prime factor.

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

BealConjecture.beal_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem beal_conjecture : bealConjecture := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Official Beal Conjecture and prize siteofficial