{"ID":2884698,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.08307","arxiv_id":"2508.08307","title":"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.","short_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 inte...","url_abs":"https://arxiv.org/abs/2508.08307","url_pdf":"https://arxiv.org/pdf/2508.08307v1","authors":"[\"Nick Craig-Wood\"]","published":"2025-08-08T11:08:13Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"cs.AI\"]","methods":"[\"LoRA\"]","has_code":false}