Detecting Primes Without Logic: A Continuous Analytic Prime Root Function

We construct a continuous and analytic function R(x), defined on the positive real numbers, whose roots occur precisely at the prime numbers. That is, R(x) = 0 if and only if x is a prime. Classical methods for detecting primes rely on discrete tools such as modular arithmetic, logic, or floor functions, which are inherently non-analytic. In contrast, our construction avoids all such operations. It is built by extending Wilson’s theorem through the Gamma function and composing it with an analytic function that evaluates to 1 if and only if its input is an integer. The result is a purely real-valued, continuous, and smooth function that encodes primality without reference to infinite series, trigonometric functions, or piecewise logic. This formulation provides a new analytic perspective on the distribution of primes and demonstrates how classical number-theoretic ideas can be embedded in continuous mathematics.