A Spectral Method for Ultra-Fast Generation of Large Prime Numbers

We present a complete numerical framework connecting the non-trivial zeros of the Riemann zeta func‐tion to the distribution of prime numbers. The work consists of three interconnected components: (i) a spectral law derived from a physical Hamiltonian whose eigenvalues approximate the non-trivial zeros of the Riemann zeta function with R² = 0.99999272, (ii) a large-scale numerical verification of Riemann's explicit formula up to 10,000 primes using 10,000 numerically computed zeros, achieving 99.9% recall and 100% precision; and (iii) a hybrid algorithm that combines the spectral law (for fast zero generation) with the explicit formula (for primality testing) to generate large primes with deterministic verification. Extensive statistical testing over 1,000 primes of 1024-bit size demonstrates 100% accuracy with mean generation time of 37.43 ms, significantly outperforming existing methods for cryptographic applications. The framework provides strong numerical evidence consistent with the Hilbert–Pólya conjecture and establishes a practical pathway for ultra-fast prime generation.