Algorithm-assisted discovery of an intrinsic order among mathematical constants

TL;DR

The paper advances a computational program to discover and organize relationships among mathematical constants by introducing conservative matrix fields, a framework that renders the space of polynomial continued fraction formulas for a constant into a path-invariant structure. It builds and deploys the Distributed Factorial Reduction algorithm to generate hundreds of new formulas, then extracts a unifying matrix-field picture that explains how different trajectory choices yield equivalent or related limits, including irrationality proofs via trajectory-dependent irrationality measures. Key contributions include a concrete CMF construction for several constants (notably $ ext{zeta}(2)$ and $ ext{zeta}(3)$), a mechanism to connect disparate constants (e.g., $ ext{pi}$, $ ext{ln}(2)$, Catalan’s constant), and a method to derive irrationality proofs from trajectory optimization. The work demonstrates the power of large-scale experimental mathematics, providing a blueprint for automatic conjecture generation, verification, and potential future proofs, while outlining open questions about uniqueness, higher dimensions, and extensions to $ ext{zeta}(5)$ and beyond.

Abstract

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Apéry for the irrationality of $ζ(3)$. Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

Figures (7)

• Figure 1: Algorithmic-driven scientific method in experimental mathematics and its realization in our work. Our work encompasses two cycles of the scientific method, leading to the discovery of a novel mathematical concept.
• Figure 2: Observation of the factorial reduction (FR) property - its connection to different constants and its sparsity in the wide space of continued fractions. (a) Calculation of the growth rate $s=\sqrt[n]{|p_n/g_n|}$ of the reduced convergents for different continued fractions, detailed in the table. We plot a negative $s$ value to denote a negative limit to $p_n/g_n$. Even very similar continued fractions can have vastly different growth rates. We observe exponential growth whenever the continued fraction converges to a mathematical constant. This approach identifies formulas for a wide range of mathematical constants. (b) Comparison of the growth rate of continued fractions of the same form shows that the only one having factorial reduction is precisely the one found by Apéry aperypoorten to converge to $\zeta(3)$. The factorial reduction property is extremely rare, shown by its sensitivity to slight changes in the parameters of the continued fraction. (c) Table of the polynomial continued fractions calculated in panel (a), denoting which ones have or do not have factorial reduction, and which ones converge to a known constant or to a constant with no known closed form. Here, $G$ is Catalan's constant, and $\hat{\zeta}(5,1)= \zeta(5)-\zeta(4)+\zeta(3)-\zeta(2)+1$ is the continued fraction from Eq. \ref{['eq:zzz_general']} with $s=5,R=1$.
• Figure 3: Implementation of the Distributed Factorial Reduction (DFR) algorithm. On-site scheme creation: Our database collects formulas from the literature and from past results of algorithms. This collection helps us define schemes of parameterized polynomial continued fractions. The range of parameters of the search space is decided based on our computational power. In this work, our search spaces are families of polynomial continued fractions. Off-site factorial reduction testing: Our BOINC resource manager server distributes the search space between the workers, who evaluate the polynomial continued fractions and test each one for factorial reduction. This step is the most computationally intensive part of the algorithm. The computing power that enables this effort is donated by volunteers from the BOINC community, and, at the time of writing, involves over 5000 personal computers. The rare events of positive identification are sent back for further analysis. On-site result verification: each identified continued fraction is first independently verified for factorial reduction. We then attempt to match each verified continued fraction to constants using PSLQ PSLQ1992PSLQ1999. Each PSLQ match is then validated to greater precision. We also collect the results for which a PSLQ match was not found, to enable future tests with additional constants, refined applications of PSLQ, and potentially superior algorithms that may emerge.
• Figure 4: Generating an efficient converging sequence from an infinite family of continued fractions. The table presents a parametric family of continued fractions. The rows correspond to integer values of $\alpha$, which provide formulas converging to linear fractional transformations of $\zeta(3)$. Inverting this transformation for every element in each sequence creates sequences that all approach $\zeta(3)$. Then, we construct a new sequence by sampling the "diagonal" trajectory along the 2-dimensional grid of sequences, i.e., we select each consequent element from the consequent sequence. This constructed sequence creates an efficient approximation of $\zeta(3)$ that proves its irrationality.
• Figure 5: Comparison of conservative vector fields and conservative matrix fields. (a) Conservative vector fields exhibit path-independence, meaning that integrals taken over different paths yield the same result. (b) Analogously, conservative matrix fields also offer path-independence, but for multiplications along the paths. The resulting matrices only depend on the initial and final positions. Interestingly, conservative matrix fields are found to exhibit additional features: paths going to infinity represent continued fraction formulas. (c) Each presented path provides a different formula of the same mathematical constant. The example continued fractions here are derived from the conservative matrix field of $e$ in Appendix D.1. Such continued fractions can be derived for any constant that has a conservative matrix field. We hypothesize that all continued fractions and infinite sums converging to a certain constant can be derived from a single conservative matrix field of that constant.