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

arXiv:2508.08307v2 Announce Type: replace-cross 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.

Source: arXiv cs.AI | 2026-09-17