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

TL;DR

This paper introduces a new method combining PSLQ algorithm with algebraic filters to discover low-Lehmer-measure Machin-like identities, resulting in record-low measures and enabling the generation of longer pi formulas.

Contribution

The paper presents a novel framework coupling PSLQ with number-theoretic filters to efficiently find low-Lehmer-measure Machin-like identities, achieving record measures and extending to longer formulas.

Findings

Discovered new 5 and 6 term identities with record-low Lehmer measures.

Developed a scalable search framework combining PSLQ with algebraic filters.

Enabled generation of longer pi formulas through algorithmic extensions.

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.