用数学约束加速搜索π的高效公式，创下最低测度纪录
Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures
- 结合PSLQ算法与高斯整数结构过滤，大幅缩小搜索空间
- 发现5项和6项公式，测度低至1.4572和1.3291，创纪录
- 方法可推广生成更长公式，适合数学计算优化研究者
Machin-like反正切恒等式是计算π的经典工具，其效率由莱默测度（λ）衡量。本文提出一种框架，通过将PSLQ整数关系算法与高斯整数代数结构导出的数论过滤器结合，使大规模搜索变得可行。搜索获得新的5项和6项恒等式，莱默测度分别为λ=1.4572和λ=1.3291，达到当前最低纪录。我们还展示了如何利用发现的恒等式作为基础，通过算法扩展生成更长的新公式。这种受限的PSLQ搜索与算法拓展相结合的方法，为未来探索提供了稳健路径。
原文摘要 · Abstract (English)
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.
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