Topological Screening Optimization System for Billion-Digit and Larger Mersenne Primes

English Abstract To solve the excessive computing redundancy and low detection efficiency caused by brute-force traversal in the GIMPS distributed prime-search project, this paper proposes an original Shui’s Topological Screening System. Based on modulo periodic sieve theory, binary generating element model and self-developed π wave phase convergence algorithm, a dedicated deterministic pre-screening mechanism for billion-digit and larger Mersenne primes is established. Verified by ten groups of billion-level exponent tests, all valid prime exponents must satisfy modulo 30 high-energy congruence constraints and binary topological steady-state characteristics. This system can be embedded into official GIMPS clients to eliminate invalid exponents and reduce global distributed computing waste. All contents are fully open-source, providing verifiable technical support for super-large prime detection and EFF Collaborative Computing Award evaluation.