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素数等差数列的最小参数计算：基于2³²范围的系统性搜索
Minimal Parameter Computation for Prime Arithmetic Progressions: A Systematic Search up to 2^32
V2
摘要
素数等差数列是数论的核心研究对象。尽管格林–陶定理断言了任意长素数等差数列的存在性，但构造具有极小参数（最小首项或最小公差）的具体实例仍是重大的计算挑战。本文提出一种系统性的二阶段搜索算法，在 2^32（约4.29×10⁹）范围内的约2.03亿个素数中展开大规模计算，旨在发现特定长度 k 的、参数极小的素数等差数列。主要工作如下：（1）在统一的 2^32 范围内，完成了对长度 k=2 至 16 的素数等差数列最小公差解与最小首项解的系统性搜索，获得了一系列最优解；（2）基于详实的计算日志，揭示了算法耗时、最小公差倍数与项数 k 的数学性质（尤其是素数性）之间的内在关联；（3）通过与作者早前较小范围内的结果对比，初步探讨了最小首项与搜索范围的关系。本研究获得的一系列最优解，为素数分布的结构特性提供了新的具体证据，并为后续建立更精确的理论模型奠定了实证基础。
Abstract
Prime arithmetic progressions are a core research subject in number theory. Although the Green–Tao theorem asserts the existence of arbitrarily long prime arithmetic progressions, constructing concrete instances with minimal parameters (minimal first term or minimal common difference) remains a significant computational challenge. This paper proposes a systematic two-stage search algorithm and conducts large-scale computation among approximately 203 million primes within the range of 2^32 (about 4.29×10^9), aiming to discover prime arithmetic progressions of specified length k with extremely small parameters. The main contributions are as follows: (1) within the unified range of 2^32, a systematic search for minimal common difference solutions and minimal first term solutions of prime arithmetic progressions with lengths k=2 to 16 is completed, yielding a series of optimal solutions; (2) based on detailed computational logs, the intrinsic correlation between algorithm runtime, the minimal common difference multiplier, and the mathematical properties of the length k (especially primality) is revealed; (3) through comparison with the author's earlier results in a smaller range, the relationship between the minimal first term and the search range is preliminarily discussed. The series of optimal solutions obtained in this study provides new concrete evidence for the structural characteristics of prime distribution and lays an empirical foundation for subsequent establishment of more precise theoretical models.关键词
素数等差数列/最小首项/最小公差/最优解/计算数论/搜索算法/格林–陶定理/基础公差Key words
prime arithmetic progression/ minimal first term/ minimal common difference/ optimal solution/ computational number theory/ search algorithm/ Green–Tao theorem/ base primorial引用本文复制引用
曾飞戈.素数等差数列的最小参数计算：基于2³²范围的系统性搜索[EB/OL].(2025-10-09)[2026-10-29].https://sinoxiv.napstic.cn/article/26423329.学科分类
数学版本历史
| 版本 | 时间 | DOI号 | 操作 |
|---|---|---|---|
| v2（当前版本） | 2026-10-29 15:55:29 | 10.12383/202510010001V2 | 全文下载 |
| v1 | 2025-10-09 10:37:59 | 10.12383/202510010001V1 | 全文下载 |