Beal Conjecture Convincing Proof

The author proves the original Beal conjecture that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. In the numerical equations, two approaches have been used to change the sum, A^x + B^y, of two powers to a single power, C^z. In one approach, the application of factorization is the main principle, while in the other approach, a formula derived from A^x + B^y was applied. The two approaches changed the sum A^x + B^y to a single power, C^z, perfectly. The derived formula confirmed the validity of the assumption that it is necessary that the sum A^x + B^y has a common prime factor before C^z can be derived. It was concluded that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor.