摘要
素数筛法始终将“直接定位素数”作为核心目标，限于“素数无显性规律”的瓶颈，只能通过优化遍历效率来缓解这一局限，无法实现本质突破。论文转变研究视角，将“无规律的素数问题”转化为“有规律的合数问题”，改变了筛法的逻辑起点；揭示了特殊合数的分布规律，提出递进推演法，构建了“已知素数→推演特殊合数→筛选素数→扩展素数集”的完整分类分段推演迭代闭环，实现了特殊合数的规律推演与素数的精准筛选，该方法具有自然迭代扩展的理论可能性；减少了传统筛法在范围覆盖、逻辑处理及计算等过程中的冗余，且运算方法更为简洁，在时空复杂度优化、迭代扩展性及工程实用性等方面显示出一定的理论优势。
关键词： 数论；素数筛法；合数分布规律；递进推演法
Abstract
The prime sieve method has always taken "directly locating prime numbers" as its core objective. However, due to the bottleneck of "prime numbers having no explicit pattern", it can only alleviate this limitation by optimizing the efficiency of traversal, without achieving an essential breakthrough. This paper shifts the research perspective, transforming the "irregular prime number problem" into the "regular composite number problem", changing the logical starting point of the sieve method; it reveals the distribution pattern of special composite numbers, proposes the progressive deduction method, and constructs a complete iterative closed loop of "known prime numbers → deducing special composite numbers → screening prime numbers → expanding the prime number set". This method realizes the regular deduction of special composite numbers and the precise screening of prime numbers, and has the theoretical possibility of natural iterative expansion. It reduces the redundancy in the process of range coverage, logical processing, and calculation in traditional sieve methods, and its operation method is more concise. It shows certain theoretical advantages in terms of time and space complexity optimization, iterative expandability, and engineering practicality.
Key words： Number theory; Prime sieve method; Distribution law of composite numbers; Progressive deduction method
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