Bahbouhi Bouchaib
This study introduces a new theoretical and computational framework for detecting prime numbers within random numerical sequences. Building upon previous models such as the Unified Prime Equation (UPE) and the Z Constant, the present work extends their predictive capacity to unstructured domains, showing that prime emergence is not a product of randomness but a manifestation of deeper harmonic regularities. By applying recurrencebased scanning and cross-correlation with normalized logarithmic spectra, the model reveals deterministic signals corresponding to prime locations in large pseudo-random sequences. The probability distribution of primes, when projected onto these harmonic coordinates, converges toward a stationary profile that remains consistent across multiple scales up to 10^12. Empirical results demonstrate that even in artificially generated random contexts, the primes are not evenly dispersed but appear in statistically constrained corridors defined by the UPE–Z interaction law. The paper also proposes an algorithmic criterion to classify numbers as potentially prime based on their spectral deviation score, which shows high accuracy and strong alignment with actual prime indices. These findings support the hypothesis that primes can be predicted from non-sequential environments, implying that randomness and determinism coexist under a unified law of distribution. This approach not only deepens our understanding of prime genesis but also contributes to ongoing discussions surrounding the deterministic nature of the Riemann zeta function and its relation to arithmetic symmetries. The results may pave the way toward a constructive understanding of prime dispersion and new avenues in probabilistic number theory and computational prediction.
Bahbouhi Bouchaib, Independent Researcher, Nantes, France.
Bahbouhi. B, (2025). The Unified Prime Equation and the Z Constant: A Constructive Path Toward the Riemann Hypothesis. Comp Intel CS & Math , 1(1), 01-33.