A Proof of Beal's Conjecture via Structural and p-adic Analysis

This manuscript provides a proof of Beal's Conjecture by contradiction. The proof assumes the existence of a non-trivial, coprime integer solution to the equation Aˣ + Bʸ = Cᶻ where x, y, z > 2 and gcd (A, B, C) = 1. The proof proceeds by eliminating all possible structures for a coprime solution: Parity analysis reduces the problem to two viable scenarios: (Odd + Even = Odd) and (Odd + Odd = Even). The (Odd + Even = Odd) case is proven impossible by reducing the equation to a form of the Generalized Catalan Equation (Cᶻ - Aˣ = 2ᵏ) which is known to have no solutions. The (Odd + Odd = Even) case is proven impossible by splitting it into two sub-cases: a. The Symmetric Case (x=y) is disproven using Zsigmondy's Theorem, which leads to a p-adic contradiction (1 >= 3). b. The Asymmetric Case (x!=y) is disproven using 2-adic (nu₂) valuation, which creates an irreconcilable contradiction between the "supplied" exponent of the LHS (nu₂ = 1) and the "required" exponent of the RHS (nu₂ >= 4). As all possible forms of a coprime solution lead to a contradiction, the initial assumption must be false. Therefore, any solution must have a common prime factor.