Unsupervised Discovery of Formulas for Mathematical Constants

TL;DR

This paper introduces a novel methodology for automatically discovering and categorizing formulas for mathematical constants by analyzing their convergence dynamics, leading to new formulas and insights into mathematical structures.

Contribution

The authors develop a convergence-based metric system enabling automated clustering of formulas, and demonstrate its effectiveness on a large dataset to find both known and new formulas for constants.

Findings

Successfully identified formulas for π, ln(2), and other constants.

Discovered previously unknown formulas for several mathematical constants.

Enabled generalization of formulas into infinite families, revealing structural patterns.

Abstract

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for $\pi$, $\ln(2)$, Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.