Fermat's General Case 4th Way

Abstract This paper presents a new approach to generalizing Fermat’s Last Theorem by introducing the Fermat’s General Case Theorem – 4th Way, which investigates the equation an+bm=cza^n + b^m = c^z under various geometric configurations and number-theoretic constraints. Using original constructs such as Taha’s Coefficient Fact (TCF1) and the Three-Sided Geometric Shapes Fact (TNGSF), the theorem explores five distinct cases—right, acute, and obtuse triangles, and two segment conditions—demonstrating that for all n,m,z∈N+n, m, z \in \mathbb{N}_+ with n>3n > 3, the equation does not hold under these configurations unless specific conditions are met. This framework offers a fresh perspective on the interplay between geometry and number theory.