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August 19, 2025Open Access

Fermat's General Case 4th Way

Puntos clave

The theorem shows that the equation an+bm=cza^n + b^m = c^z is not valid under most geometric conditions.
It establishes that for all n,m,z∈N with n>3, specific configurations must be present for the equation to hold.
Utilizing concepts like Taha’s Coefficient Fact and Geometric Shapes Fact, it explores various triangle types.
The findings emphasize the connection between geometry and number theory, suggesting new mathematical pathways.

Resumen

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=czaⁿ + bᵐ = cᶻ 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 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.

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