Sum of Powers Applied in Cartesian Plane

This paper proposes a demonstration that if the sum of powers where A^x and B^y have a common prime factor, and generates a compound number C^z, then Beal s conjecture involving A^x+ B^y=C^z is true for all A, B , C, x, y, z positive integers and x, y, z 2, as well as showing types where the sum A^x + B^y generates no prime factor among A, B and C.