From Euler to AI: Unifying Formulas for Mathematical Constants
Tomer Raz, Michael Shalyt, Elyasheev Leibtag, Rotem Kalisch, Shachar Weinbaum, Yaron Hadad, Ido Kaminer
The constant $\large \pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many 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 $\large \pi$, an ideal testing ground for symbolic unification. Applying this approach to 455,050 arXiv papers, we validate 407 distinct formulas for $\large \pi$ and prove relations between 381 (94\%) of them, of which 188 (46\%) 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 $e$, $\zeta(3)$, and Catalan’s constant, demonstrating the potential of AI-assisted mathematics to uncover hidden structure and unify knowledge across domains.