From Euler to AI: Unifying Formulas for Mathematical Constants

The constant $\pi$ has fascinated scholars for centuries, inspiring the derivation of countless formulas rooted in profound mathematical insight. This abundance of formulas raises a question: Are they interconnected, and can a unifying structure explain their relationships? We propose a systematic methodology for discovering and proving formula equivalences, leveraging modern large language models, large-scale data processing, and novel mathematical algorithms. Analyzing 457,145 arXiv papers, over a third of the validated formulas for $\pi$ were proven to be derivable from a single mathematical object - including formulas by Euler, Gauss, Lord Brouncker, and newer ones from algorithmic discoveries by the Ramanujan Machine. Our approach extends to other constants, such as $e$, $\zeta(3)$, and Catalan's constant, proving its broad applicability. This work represents a step toward the automatic unification of mathematical knowledge, laying a foundation for AI-driven discoveries of connections across scientific domains.