Treatise on Circles through Natural Numbers

This treatise introduces Prime-Diameter Circles (PDCs)—a geometric framework that unifies prime generation, distribution, and classical number-theoretic conjectures. By defining circles whose diameters are bounded by prime coordinates, PDCs reveal profound symmetries: primality reflection, parity constraints, and digital index restrictions. These properties naturally encode twin primes, prime gaps, and arithmetic progressions, while offering geometric restatements of celebrated conjectures. Extending the framework, Factor-Finder Circles provide geometric divisibility and primality tests, bridging continuous geometry with discrete number theory. Together, these constructions demonstrate that primes, often studied through analytic or algebraic lenses, can be elegantly characterised through circle symmetries—opening new pathways for visualising, approximating, and testing primality. This work highlights the profound mathematical unity achieved when prime theory is reframed through geometry.