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.
Ricardo Adonis Caraccioli Abrego (Fri,) studied this question.