Prime Recurrence in Circular Multi-Reflection Optical Systems

This work investigates a physical model in which prime numbers emerge as recurrence conditions in a circular multi-reflection optical system. Mirrors are arranged at the vertices of a regular polygon, with one half-mirror and the remaining mirrors fully reflective. The system exhibits a return of light to the initial point only when the number of mirrors is prime. We interpret this phenomenon in terms of cyclic groups, primitive roots, and wave interference, suggesting a possible physical realization of number-theoretic structures. Connections to the distribution of primes, the appearance of π in analytic number theory, and potential computational applications are discussed.