Beal's Conjecture

Claim: if \(A^{x}\) + \(B^{y}\) = \(C^{z}\) holds in positive integers with min(x,y,z) ≥ 3, then A, B, C share a common prime factor.

Formulated by Dallas banker Andrew Beal in 1993 while investigating Fermat patterns; he raised the prize to $1,000,000 (administered by the American Mathematical Society) in 2013. It generalises FLT and sits inside the Fermat–Catalan programme: Tijdeman's framework shows fixed exponent triples yield finitely many primitive solutions (several fully dispatched — e.g. x=y=z=3 is FLT itself; (2,3,7)-type equations resolved via modularity and hyperelliptic methods). The uniform statement resists everything thrown at it.