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

Abstract

Machin-like arctangent relations are classical tools for computing $π$, with efficiency quantified by the Lehmer measure ($λ$). 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 ($λ=1.4572, λ=1.3291$). 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.