Diophantus' On Polygonal Numbers via Recursive Fractal Dynamics and Prime-Modulated Geometry

We rigorously complete Diophantus’ seminal work, On Polygonal Numbers, by synthesizing classical number-theoretic results with modern analytic, algebraic, and computational techniques. This synthesis extends polygonal number theory through recursive solutions of Pell equations, analytic connections to zeta functions and modular forms, and prime-modulated torsion fields. By incorporating fractal dynamics and geometric deformations, we establish a comprehensive framework for polygonal representations of integers, revealing new structural symmetries and density theorems within the space of polygonal partitions