A Precision Prime Number Generator Based on Modular Distribution Statistics: From 8-Orbit Structure to Convergence Coefficient Locking

This paper presents a novel prime number generation method—a sieve based on modular distribution statistics. The method originates from an elementary number theory fact: all primes greater than 5 necessarily fall into the 8 residue classes modulo 30, which determine arithmetic progressions (termed the 8 orbits). We leverage the convergence property of squaring to generate candidates, and perform stepwise sieving by counting residue frequencies and applying a dynamic threshold (T = max (counts) /2 + δ), where δ is the convergence coefficient. Through algorithmic optimization, we have achieved fully automatic sieving and locked the convergence coefficient at (δ = 0. 5). Experiments show that for (N=10⁸), with (δ = 0. 5) the program yields a candidate set with prime proportion 59%, continuous up to (311²) (approximately 96703) in about 120 seconds, and the set reaches a "prime + semiprime" binary pure state (other composites are 0. 18%). Further verification reveals that the convergence coefficient δ can vary within a wide interval (0. 5, 10) without affecting the final candidate set size, demonstrating the high robustness of the sieve. Independent verification of the 8 orbits shows that except for orbit 13, which slightly deviates due to natural fluctuations, the prime proportions in the remaining orbits are highly consistent (59. 13%–59. 15%), semiprime proportions are (40. 67%–40. 71%), and other composites are below 0. 18%, fully proving the uniformity and stability of the sieve. Compared with the classical Sieve of Eratosthenes, our method requires no precomputed prime table, and its computational load decreases as the candidate set shrinks. An analysis of the last-digit structure of RSA moduli is conducted using the orbit classification. Finally, we discuss the method's potential application on quantum computers, as its probabilistic nature aligns naturally with quantum parallelism.