Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures

Abstract:Machin-like arctangent relations are classical tools for computing $\pi$, with efficiency quantified by the Lehmer measure ($\lambda$). We present a framework for discovering low-measure relations by coupling the PSLQ integer-relation algorithm with number-theoretic filters derived from the algebraic structure of Gaussian integers, making large scale search tractable. Our search yields new 5 and 6 term relations with record-low Lehmer measures ($\lambda=1.4572, \lambda=1.3291$). We also demonstrate how discovered relations can serve as a basis for generating new, longer formulae through algorithmic extensions. This combined approach of a constrained PSLQ search and algorithmic extension provides a robust method for future explorations.

26 pages, 2 tables. v2: corrects the previously best known 5, 6 and 9 term relations (Section 1.2, Table 1) and attributes the floor-function iteration to Abrarov et al. (Section 3.3). Results unchanged

Number Theory (math.NT); Artificial Intelligence (cs.AI)

arXiv:2508.08307 [math.NT]

(or arXiv:2508.08307v2 [math.NT] for this version)