Unified Approaches to Congruent Numbers via Geometry, Elliptic Curves and Arithmetic Progression of Squares

We explore congruent numbers through a unified approach combining geometric constructions, elliptic curve theory, and arithmetic progressions of squares. Leveraging recent advances in modular forms and computational techniques, we construct infinite explicit families of congruent numbers parameterized by Pell-type equations and related Diophantine conditions. We complement this with a detailed statistical analysis of the residue classes, rank distributions, and root numbers associated with these families, providing empirical insights that deepen understanding of intricate conjectures in the arithmetic of elliptic curves.