A Dual-Perspective Representation of Primes under the Safe Residue Class Sieve Framework: Local Static Characterization and Global Recursive Generation安全剩余类筛法框架下素数的双视角表示:局部静态刻画与全局递推生成

This paper presents explicit formulas for prime parameters based on the Safe Residue Class Sieve under the Modulo-6 Framework. Two types of formulas are established. The first is a local static formula: for a fixed finite sieving set, the k-th prime parameter within the first complete period is expressed via a mixed-radix expansion of k-1 combined with the Chinese Remainder Theorem. The second is a global dynamic generation formula: the entire sequence of prime parameters is generated by a deterministic finite automaton whose state transitions are governed by the forbidden residue classes of the sieves. Specifically, if r_k is the current prime parameter, the next prime parameter is given by r_{k+1} = r_k + d(a1, ..., am), where d is the minimal positive integer such that r_k + d avoids all forbidden residue conditions. The jump distances take values in {1, 2, ..., max q_i} and depend only on the current residue vector of r_k modulo each sieve modulus. This yields a closed-form recurrence that generates all safe residue class constant terms, hence all prime parameters. Numerical examples are provided to verify both formulas. The entire derivation relies solely on the Safe Residue Class Sieve and the Chinese Remainder Theorem.

本文在模六框架与安全剩余类筛法的基础上,建立素数参数的显式公式。本文给出两类公式。第一类是局域静态公式:对于固定的有限筛子集,第一个完整周期内的第k个素数参数通过对k-1进行变进制展开并结合中国剩余定理合成而显式表达。第二类是全局动态生成公式:全体素数参数序列由一个确定性有限自动机生成,其状态转移由各筛子的禁止剩余类控制。具体地,若当前素数参数为r_k,则下一个素数参数为r_{k+1} = r_k + d(a1, ..., am),其中d是使得r_k + d避开所有禁止同余条件的最小正整数。跳跃距离取值于{1, 2, ..., max q_i},且仅依赖于r_k在各筛子模数下的当前余数向量。这给出了生成全体安全剩余类常数项、从而生成全体素数参数的一个封闭递推公式。本文给出数值示例验证两类公式的正确性。整个推导仅依赖安全剩余类筛法和中国剩余定理。