# From Euler to AI: Unifying Formulas for Mathematical Constants

arXiv: https://arxiv.org/abs/2502.17533 · HTML: https://mlsanity.com/paper/2502.17533/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 95 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Builds on (most cited first, 6 of 6)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants](https://mlsanity.com/paper/1907.00205/) (2019)
- [Algorithm-assisted discovery of an intrinsic order among mathematical constants](https://mlsanity.com/paper/2308.11829/) (2023)
- [Automated Search for Conjectures on Mathematical Constants using Analysis of Integer Sequences](https://mlsanity.com/paper/2212.09470/) (2022)
- [Unsupervised Discovery of Formulas for Mathematical Constants](https://mlsanity.com/paper/2412.16818/) (2024)

## Built on by (most cited first, 2 of 2)

- [Not All Proofs Are Equal: Evaluating LLM Proof Quality Beyond Correctness](https://mlsanity.com/paper/2605.10379/) (2026)
- [Can AI Follow In Einstein's Footsteps?](https://mlsanity.com/paper/2607.27794/) (2026)

## Abstract (arXiv)

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, zeta(3), and Catalan's constant, demonstrating the potential of AI-assisted mathematics to uncover hidden structures and unify knowledge across domains.