The Fractal Geometry of Numbers: A Geometric Construction of Integers q • 2 k and Their Primality Structure

This paper presents a novel geometric framework that constructs integers of the form q • 2 k (with q odd) through a recursive binary subdivision of regular polygons inscribed in the unit circle. The construction reveals a fractal hierarchy of triangles and lunules-a "Russian doll" structure-where each level corresponds to multiplication by 2. We prove that the odd factor q remains invariant under this binary subdivision, providing a geometric interpretation of the factorization n = q • 2 a . The framework yields several applications: (1) a visual characterization of primality through the absence of inscribed regular subpolygons, (2) a geometric interpretation of classical primality tests (Lucas-Lehmer for Mersenne numbers 2 p -1 and Proth's theorem for k • 2 n + 1), and (3) a fractal "microscope" that visualizes divisor structure. We establish explicit bijections between arithmetic progressions and geometric configurations, linking dihedral symmetry to divisibility. The approach is both theoretically rigorous and pedagogically illuminating, offering fresh insights into the interplay between number theory and geometry.