arXiv:2502.17533math.HOcs.AI2025-02NeurIPS被引 4

用AI统一数学常数公式,发现π的360个公式可归一为同一对象。

From Euler to AI: Unifying Formulas for Mathematical Constants

  • 结合大模型与符号算法,自动挖掘并验证数学公式关联
  • 从45万论文中验证385个π公式,94%揭示内在联系
  • 首次将欧拉、高斯等经典公式与现代算法成果统一

π自古以来吸引学者关注,催生了大量求值公式,如无穷级数与连分数。然而,这些公式的深层关联长期不明,缺乏统一理论。本文提出一种自动化框架,融合大语言模型系统性收集公式、代码反馈循环验证,并设计新型符号算法实现聚类与统一。以π为典型测试对象,分析455,050篇arXiv论文,验证385个π公式,证明其中360个(94%)存在关联,且166个(43%)可由单一数学对象导出,涵盖欧拉、高斯、布朗克及拉马努金机器算法发现的公式。该方法可推广至e、ζ(3)和卡塔兰常数,展现人工智能辅助数学在揭示隐藏结构、统合跨领域知识方面的潜力。

原文摘要 · Abstract (English)

The constant $π$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections among formulas remain unknown, missing unifying theories that could unveil deeper understanding. The absence of a unifying theory reflects a broader challenge across math and science: knowledge is typically accumulated through isolated discoveries, while deeper connections often remain hidden. In this work, we present an automated framework for the unification of mathematical formulas. Our system combines Large Language Models (LLMs) for systematic formula harvesting, an LLM-code feedback loop for validation, and a novel symbolic algorithm for clustering and eventual unification. We demonstrate this methodology on the hallmark case of $π$, an ideal testing ground for symbolic unification. Applying this approach to 455,050 arXiv papers, we validate 385 distinct formulas for $π$ and prove relations between 360 (94%) of them, of which 166 (43%) can be derived from a single mathematical object - linking canonical formulas by Euler, Gauss, Brouncker, and newer ones from algorithmic discoveries by the Ramanujan Machine. Our method generalizes to other constants, including $e$, $ζ(3)$, and Catalan's constant, demonstrating the potential of AI-assisted mathematics to uncover hidden structures and unify knowledge across domains.

AI数学常数统一符号计算

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