From Euler to AI: Unifying Formulas for Mathematical Constants

Abstract: Mathematical constants have fascinated scientists for thousands of years. Countless formulas for constants like pi, e, zeta(3), and Catalan’s constant have accumulated over time, yet no centralized theory explaining all of them exists. In this talk, we will dive into the workings of a fully automated system developed for harvesting, validating, and unifying formulas for mathematical constants. Enabling the unification of formulas is one of the first successful hybrid LLM-symbolic tool systems. This system constitutes the first application of LLMs to number theory, a topic that has remained largely elusive to AI. The talk will review some of the key milestones in AI for math and achievements in number theory in particular. We will explain what roles LLMs have played so far, and how they are enhanced by symbolic tools: In our unification system, formulas are harvested automatically by LLMs from hundreds of thousands of arXiv publications, validated numerically, and fed into a novel symbolic algorithm. The joint system thus integrates the LLM’s flexibility in NLP of mathematical expressions with the rigor of symbolic algorithms. As an example result, we will show that many formulas for pi, including famous ones by Euler, Gauss, and the Ramanujan Machine, are generated by a common mechanism. We will further discuss what steps are necessary to complete the journey and unify all formulas for pi and other mathematical constants. As a preview of ongoing work, we will present preliminary results from a hybrid LLM-symbolic tool built to achieve record integral solving capabilities.