Computation of Minimal Parameters for Prime Arithmetic Progressions: A Systematic Search within 2³²

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.