Optimization of Distributed Prime Search via Symmetric Primorial Recursion (SPR) Framework: Candidate Density Optimality and Energy Efficiency Analysis in GIMPS and PrimeGrid Projects

Distributed prime searching represents one of the most challenging intersections of computational number theory and distributed computing. The Great Internet Mersenne Prime Search (GIMPS), initiated by George Woltman in 1996, and the BOINC-based PrimeGrid platform have collectively discovered numerous theoretically significant and practically valuable special primes. On October 21, 2024, GIMPS announced the discovery of the 52nd Mersenne prime (41,024,320 decimal digits), marking the end of the 28-year era of ordinary personal computers finding the largest known primes, as Luke Durant utilized cloud-based GPU clusters spanning 17 countries to achieve this breakthrough . However, traditional search methods still rely on random traversal strategies for candidate exponent generation, leaving substantial room for theoretical efficiency improvements. This paper proposes a unified optimal candidate generation method based on the Symmetric Primorial Recursion (SPR) framework. The SPR recurrence is defined as: with the convention that symmetric elements are automatically included. This recursion generates the complete reduced residue system modulo , and elements of in the interval are precisely the primes. Theoretical analysis demonstrates that the prime density of SPR-generated candidates reaches , approximately 12.3% higher than random odd prime density, achieving 88% of the information-theoretic lower bound. Combined with divisor structure pre-screening, SPR reduces candidates entering Lucas-Lehmer or LLR testing to 44.5% of traditional methods, effectively halving the expected time for discovering new primes. Energy analysis indicates that had GIMPS adopted SPR methodology historically, cumulative energy consumption could have been reduced by approximately 55.5%, saving 633 million kWh and preventing 3.6 million tons of CO₂ emissions—aligning with the 2025 emphasis on green supercomputing and energy efficiency in AI data centers . This paper systematically reviews breakthroughs from 2024–2026, including the cloud supercomputing paradigm shift, PrimeGrid’s record-breaking 737 mega-primes in 2025 (averaging over 2 per day) , the first-ever GFN-21 prime discovery , and advances in primorial sieve optimization via GPU acceleration achieving 70× efficiency gains . From the SPR theoretical perspective, we analyze the limitations and complementarities of these methods, and detail SPR framework applicability across GIMPS and PrimeGrid subprojects (Sophie Germain primes, 321 primes, Cullen-Woodall primes, Extended Sierpiński Problem, Riesel Problem, Generalized Fermat primes, Wall–Sun–Sun primes, and Wieferich primes). This work provides theoretical optimization foundations and green computing value assessments for future large-scale distributed prime searches.