Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913985468563456 |
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| author | Craig-Wood, Nick |
| author_facet | Craig-Wood, Nick |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures Craig-Wood, Nick Number Theory Artificial Intelligence 11Y60 (Primary), 11Y16 (Primary), 33B10 (Secondary), 68W30 (Secondary), 11-04 (Secondary) 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. |
| title | Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures |
| topic | Number Theory Artificial Intelligence 11Y60 (Primary), 11Y16 (Primary), 33B10 (Secondary), 68W30 (Secondary), 11-04 (Secondary) |
| url | https://arxiv.org/abs/2508.08307 |