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素数等差数列的最小参数计算与若干新序列的发现
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摘要
摘要
素数等差数列（AP）是数论的核心研究对象。尽管格林-陶定理断言了任意长AP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。本文提出一种新颖的系统性搜索算法，在 π(5×) 范围内的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。主要发现如下：（1）刷新了短序列的最小首项纪录：找到了首项低至11的8项与9项AP，以及首项为19的10项AP；（2）实现了中等长度序列的突破：首次构造出首项为503,213的14项AP；（3）发现了长序列的极小公差解：获得了公差为9,699,690的16项AP，该值仅为理论最小基础公差的323倍，是当前已知的该长度最小公差解。本研究共确立了15项最优解，为素数分布的结构性提供了新的具体证据。
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
Abstract
Description Prime arithmetic progressions (APs) are a core topic of study in number theory. Although the Green-Tao theorem affirms the existence of APs of any arbitrary length, constructing specific instances with minimal parameters—specifically, the minimal starting prime or the minimal common difference—remains a significant computational challenge. This paper introduces a novel systematic search algorithm and conducts large-scale computations within the range of π(5×10⁸) primes, aiming to discover prime APs of specific length k with extremely small parameters. The main findings are as follows: (1) The minimal starting prime record for short sequences has been refreshed, uncovering 8-term and 9-term APs with a starting prime as low as 11, and a 10-term AP starting at 19; (2) A breakthrough was achieved for medium-length sequences by constructing the first 14-term AP with a starting prime of 503,213; (3) An extremely small common difference solution was found for long sequences, obtaining a 16-term AP with a common difference of 9,699,690, which is only 323 times the theoretical minimal primorial and is currently the smallest known common difference for that length. This study establishes 15 optimal records in total, providing new concrete evidence for the structural properties of prime distribution. Keywords： Prime Arithmetic Progression; Minimal Starting Prime; Minimal Common Difference; Optimal Solution; Computational Number Theory; Search Algorithm; Green-Tao Theorem; Primorial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 Starting Prime/ Minimal Common Difference/ Optimal Solution/ Computational Number Theory/ Search Algorithm/ Green-Tao Theorem/ 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与若干新序列的发现[EB/OL].(2025-10-09)[2026-10-30].https://sinoxiv.napstic.cn/article/25235190.学科分类
数学版本历史
| 版本 | 时间 | DOI号 | 操作 |
|---|---|---|---|
| v2 | 2026-10-29 15:55:29 | 10.12383/202510010001V2 | 全文下载 |
| v1（当前版本） | 2025-10-09 10:37:59 | 10.12383/202510010001V1 | 全文下载 |