2026-10-14T11:00:13Z https://www.scidb.cn/oai

10.57760/sciencedb.29083 2026-08-11T17:28:13Z

2026-08-11 Computation of Minimal Parameters for Prime Arithmetic Progressions: A Systematic Search within 2³² doi:10.57760/sciencedb.29083 en Prime arithmetic progressions (APs) are a central topic in number theory. Although the Green–Tao theorem asserts the existence of arbitrarily long APs of primes, constructing explicit instances with minimal parameters—namely, the smallest possible initial term or the smallest common difference—remains a major computational challenge. In this paper, we propose a systematic two-stage search algorithm and carry out a large-scale computation over the 200 million primes up to 2^32 (approximately 4.29×10⁹), aiming to find prime APs with extremely small parameters for given lengths k. The main contributions are: (1) a unified and complete search for both the minimal-difference APs and the minimal-start APs for lengths k=2 through 16 within the 2^32 range, yielding a precise set of benchmark data; (2) based on detailed computational logs, we reveal the intrinsic relationship among runtime, the multiplier of the minimal difference, and the number-theoretic nature of k (particularly primality); (3) by comparing with the author’s earlier results in a smaller range, we provide a preliminary discussion on the dependence of the minimal starting prime on the search bound. The fifteen optimal APs obtained in this work offer new concrete evidence for the structural distribution of primes and lay an empirical foundation for building more precise theoretical models in the future. prime arithmetic progressions; minimal initial term; minimal common difference; optimal solutions; computational number theory; search algorithm; Green–Tao theorem; base difference Zeng Feige PUBLIC https://creativecommons.org/licenses/by/4.0/ dataset Science Data Bank