Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Euler to AI: Unifying Formulas for Mathematical Constants

Tomer Raz, Michael Shalyt, Elyasheev Leibtag, Rotem Kalisch, Shachar Weinbaum, Yaron Hadad, Ido Kaminer

TL;DR

This paper tackles the long-standing challenge of unifying mathematical formulas for constants by introducing an automated framework that combines LLM-driven harvesting, LLM-code feedback loops, and a novel symbolic unification algorithm based on Conservative Matrix Fields (CMFs) and coboundary equivalence. Applied to $oldsymbol{\,\pi}$, the pipeline extracts hundreds of formulas, canonicalizes them as recurrences, and proves many equivalences, revealing that a large subset resides on a single CMF. The approach generalizes to other constants such as $oldsymbol{e}$, $oldsymbol{\zeta(3)}$, and Catalan’s constant, highlighting AI-assisted mathematics as a scalable path to uncover hidden structure across domains. These results demonstrate a scalable, data-driven route to unify mathematical knowledge and suggest future extensions to higher CMF dimensions and broader classes of constants with potential impact in both theory and computation.

Abstract

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.

From Euler to AI: Unifying Formulas for Mathematical Constants

TL;DR

This paper tackles the long-standing challenge of unifying mathematical formulas for constants by introducing an automated framework that combines LLM-driven harvesting, LLM-code feedback loops, and a novel symbolic unification algorithm based on Conservative Matrix Fields (CMFs) and coboundary equivalence. Applied to , the pipeline extracts hundreds of formulas, canonicalizes them as recurrences, and proves many equivalences, revealing that a large subset resides on a single CMF. The approach generalizes to other constants such as , , and Catalan’s constant, highlighting AI-assisted mathematics as a scalable path to uncover hidden structure across domains. These results demonstrate a scalable, data-driven route to unify mathematical knowledge and suggest future extensions to higher CMF dimensions and broader classes of constants with potential impact in both theory and computation.

Abstract

The constant 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 , an ideal testing ground for symbolic unification. Applying this approach to 455,050 arXiv papers, we validate 385 distinct formulas for 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 , , and Catalan's constant, demonstrating the potential of AI-assisted mathematics to uncover hidden structures and unify knowledge across domains.

Paper Structure

This paper contains 85 sections, 6 theorems, 205 equations, 8 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Recurrences as universal representations of formulas for mathematical constants
  4. The dynamical metrics describing each formula
  5. Conservative Matrix Fields (CMFs)
  6. Methodology for symbolic unification of formulas
  7. Methodology for symbolic unification of formulas
  8. Harvesting: large-scale retrieval of formulas from the literature
  9. Harvesting: extraction and validation
  10. Clustering: using the canonical form
  11. Clustering: using the dynamical metrics
  12. Unification: using the UMAPS algorithm for coboundary equivalence
  13. Benchmarking
  14. Comparison to other methods for symbolic unification
  15. Comparison of LLM model performance
  16. ...and 70 more sections

Key Result

Lemma 1

(A necessary condition on the coboundary equivalence matrix.) Let $L_A = \lim_{n\to \infty} PCF\left( a(n),b(n) \right) \text{ and } L_B = \lim_{n\to \infty} PCF\left( c(n),d(n) \right) \text{ be \space converging \space PCFs}$ with associated companion matrices $A(n),B(n)\in \operatorname{PGL}_2\l

Figures (8)

  • Figure 1: Selected $\pi$ formulas across centuries.
  • Figure 2: Automated methodology for unifying mathematical knowledge. A large corpus of mathematical formulas is harvested, retrieving formulas that are each translated to executable code for validation. The formulas are then clustered by conversion into their canonical forms, and unified using a novel symbolic computational algorithm that proves their relations.
  • Figure 3: Pipeline for automated harvesting of mathematical formulas (left), exemplified using one of the analyzed $\pi$ formulas (right). (a) Equations are scraped from papers on arXiv. (b) Regular expressions on the $\text{\LaTeX }$ strings retrieve series and continued fraction patterns that contain $\pi$ as the only irrational number (see \ref{['appendix-engineering-formula-patterns']}). (c) Zero-shot classification using OpenAI's GPT-4o mini identifies formulas calculating the constant $\pi$. Then, OpenAI's GPT-4o identifies the formula type (series, continued fraction, or neither). (d) Extraction of the series' summand or the continued fraction's partial numerator and partial denominator, using GPT-4o. The formula is then converted to code. (e) Formulas are computed and validated using the integer relation finder algorithm PSLQ. (f) The formulas are converted to canonical recurrences using RISC's tool for fitting recurrences kauers2022guessing.
  • Figure 4: The matching algorithm: connecting polynomial linear recurrences. This algorithm is demonstrated here for polynomial continued fractions (PCFs) but can be generalized to any linear polynomial recurrence. (a) Compute the dynamical metrics BlindDelta for the two PCFs (irrationality measures $\delta_A$, $\delta_B$ and the convergence rates ratio $r_A/r_B$). The $\delta$ metrics are used to identify possible connections, as only if $\delta_A = \delta_B$, the PCFs can be related via coboundary (in practice, we test for them to be within $0.06$ of each other). (b) Fold$\mathrm{PCF}_A$ by $r_B$ and $\mathrm{PCF}_B$ by $r_A$ (\ref{['appendix-maths-fold']}). UMAPS (c)-(e): (c) Solve for a general Möbius transform (a $2\times2$ matrix $U(1)$) that once applied to the limit of $\mathrm{PCF}_B$ equates it to the limit of $\mathrm{PCF}_A$. (d) Representing the PCFs in matrix form ($A (n)$ and $B (n)$), propagate the coboundary matrix via the relation $U(n+1) = A(n)^{-1} \cdot U(n) \cdot B(n)$ up to $U(N)$ ($N=40$ was sufficient for our runs, see \ref{['appendix-sensitivity-study']}). (e) Assume the general form of $U(n)$ to have rational-function entries with polynomial degree up to $\left\lfloor \frac{N-1}{2} \right\rfloor$ and solve for their coefficients using normalized $U(1, \ldots ,N)$. If such a solution is found and validated, the PCFs are coboundary-related. See \ref{['appendix-algs']} for more details.
  • Figure 5: Coboundary equivalence: the mathematical framework connecting different formulas once cast into their canonical forms. (a) The coboundary condition $A(n) \cdot U(n+1) = U(n) \cdot B(n)$ recasts formulas as (b,c) parallel trajectories in a CMF. (d) Example of two coboundary-equivalent formulas, presenting their coboundary matrices and limits, which constitute proof of a novel equivalence.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 1
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof
  • Lemma 4
  • proof
  • ...and 3 more