Treatise on Prime-Diameter Circles

Abstract

Jeyananth’s Circles, also called Prime-Diameter Circles (PDCs), are a geometric framework that connects prime numbers to circle constructions on the 2-D plane. Defined through prime endpoints of diameters aligned with the axes, PDCs introduce notions such as floor and ceiling coordinates, perfect and unit PDCs, and reveal properties of symmetry, parity, and digital indices. Their behaviour links naturally to arithmetic and geometric progressions, while also inspiring algorithmic approaches to prime approximation and factorisation. This treatise presents the foundations, properties, and observations of PDCs, explores their relation to progressions, and proposes tools such as a prime approximation algorithm and the Factor-Finder Circle, while leaving open questions that invite further study into the deeper structure of primes through geometry.