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

arXiv: https://arxiv.org/abs/2508.08307 · HTML: https://mlsanity.com/paper/2508.08307/

## Key facts

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- First posted on arXiv 8 Aug 2025.

## Abstract (arXiv)

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.