Circles through Natural Numbers
Year: 2026
Type: preprint
Abstract: This paper 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.
Author: Puvanenthirarasan, Ananthashayan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222800170
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