From Thousands to One: Over 99% Reduction of Mersenne Prime Candidates Through a Multi-Stage Filtering Pipeline — Stanford

We present an empirical demonstration of a unified, three-stage filtering pipelinedesigned to accelerate the discovery of Mersenne primes (Mp = 2p − 1) by drastically reducing the number of candidates requiring costly deterministic testing. Thepipeline integrates three complementary methodologies: (1) algebraic pre-screeningfilters (H2 + H1b) that eliminate ∼ 85% of composite candidates prior to primarytesting, (2) a structural priority ranking framework (S4 +S7) based on smoothnesssignatures and small order exclusions that filters the top survivors by 99%, and (3)a Lucas-Lehmer (LL) bypass utilizing Probable Prime (PRP) tests with Gerbiczerror-checking to eliminate non-PRP composites. We validate the pipeline on twocomplete, historical prime exponent gaps: M_23 → M_24 (903 prime exponents) andM_29 →M_30 (1,765 prime exponents). In both cases, the pipeline achieves an overallcandidate reduction exceeding 99.99%, systematically isolating the exact Mersenneprimes (p = 19,937 and p = 132,049) as the sole top-ranked candidates prior tofull LL validation. The process is provably safe with zero false negatives, scaleinvariant, and entirely reproducible via open-source code. This paper serves as apractical implementation guide for applying the pipeline to active search ranges,including M_53 and beyond.