Treffer: Proof of Beal's Conjecture Cuboid-Valuation Method

We present an elementary, contradiction-based proof of Beal’s Conjecture, demonstrating that the exponential Diophantine equation A^x +B^y =C^z, with pairwise coprime integers A,B,C > 1 and exponents x,y,z > 2, admits no nontrivial solutions. Our approach begins with a geometric cuboid construction that reformulates the volume identity as a partition of a large cuboid into subcuboids. We then analyze all possible exponent patterns through four exhaustive cases: 1. A Fermat-type contradiction ruling out gcd(x,y,z) > 1 directly, implying gcd(x,y,z) = 1 without analytic bounds. 2. A gap argument trapping the common edge-length between consecutive integers when z | x or z | y. 3. Aprimitive-divisor & LTE argument using Zsigmondy’s theorem for differences when gcd(z,x) > 1 or gcd(z,y) > 1. 4. A sum-of-powers Zsigmondy/LTE argument establishing pairwise coprimality of x,y,z. Combining these methods completes an unconditional proof of Beal’s Conjecture.