A Priority Ranking Framework for Mersenne Prime Candidates: Optimizing Lucas-Lehmer and PRP Testing — Stanford

After initial sieving (e.g., trial division and P-1 factoring), a large number of candidate exponents remain for Lucas-Lehmer or PRP testing. We present a lightweight priority ranking framework, R(p), designed to sit between these sieves and the final deterministic tests. Candidates with S_7(p) = 0 are purged using a proven necessary condition; survivors are ranked by S_4(p), a smoothness-based heuristic. Validated on all known Mersenne primes up to M_31, the framework consistently identifies the next Mersenne prime within the top 5 candidates in a 5× search range. A built-in safety net guarantees that no candidate is ever lost, and the total number of tests never exceeds the brute-force baseline. This framework drastically improves resource allocation for distributed computing projects such as GIMPS.