A Pedagogical Analytic Reformulation of Wilson and Fermat Prime Detectors

This note presents a pedagogical analytic reformulation of two classical primality criteria: Wilson's theorem and Fermat's little theorem in base 2. Rather than proposing a new practical primality test, we reinterpret these criteria through elementary quotient constructions and associated trigonometric indicator functions. In the Wilson setting, we obtain an exact two-branch decomposition where the quotient ((n-1)!+1)/n equals an integer if n is prime, and an integer plus 1/n if n is composite (for n > 4). This leads to a normalized analytic prime indicator. In the Fermat setting, the corresponding construction detects the set of integers that satisfy the base-2 Fermat congruence, thereby including odd primes and base-2 pseudoprimes. The main purpose of the paper is conceptual and pedagogical: to show how historical, discrete congruence criteria can be recast as exact analytic objects that help students and readers connect modular arithmetic, prime detection, and continuous or trigonometric formulations.