The Abdeslam Prime Equation: A Symbolic Framework for Deterministic Prime Construction

This paper introduces the Abdeslam Prime Equation (APE), a symbolic framework for constructing prime numbers through deterministic modular arithmetic. Unlike traditional methods that rely on randomness or probabilistic testing, APE defines primes using structured residue classes and symbolic exponents. This reveals that primes follow predictable, lawful patterns rather than appearing at random. APE enables both the forward construction of new primes and reverse validation of known ones using symbolic certificates. These certificates provide full traceability and eliminate the need for probabilistic primality tests. Validated across small and large prime sizes—including cryptographic standards—APE supports deterministic RSA key generation and offers a structured foundation for post-quantum cryptographic systems. Concepts such as residue density and adaptive modulus selection further enhance its efficiency. By transforming prime generation into a symbolic process, the Abdeslam Prime Equation offers a new perspective on number theory and introduces a deterministic alternative to conventional prime discovery.