Computing Minimal Parameters and Discovering New Sequences of Prime Arithmetic Progressions

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