Beal conjecture proof

This paper presents a complete resolution of the Beal Conjecture, which proposes that any solution to the equation A^x + B^y = C^z with positive integers A, B, C and exponents x, y, z all greater than 2 must have a common prime factor among A, B, and C. Through a combination of prime factorization analysis, modular arithmetic, and bounding arguments, the proof demonstrates that any counterexample leads to contradictions in number theoretic structure. The result generalizes and extends techniques related to Fermat’s Last Theorem, affirming the conjecture’s validity.