Algorithm-assisted discovery of an intrinsic order among mathematical constants

by   Rotem Elimelech, et al.

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.


page 15

page 30


The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants

Fundamental mathematical constants like e and π are ubiquitous in divers...

Mathematical Proof Between Generations

A proof is one of the most important concepts of mathematics. However, t...

Homogeneous length functions on Groups: Intertwined computer & human proofs

We describe a case of an interplay between human and computer proving wh...

SAT Solvers and Computer Algebra Systems: A Powerful Combination for Mathematics

Over the last few decades, many distinct lines of research aimed at auto...

The Elfe System - Verifying mathematical proofs of undergraduate students

Elfe is an interactive system for teaching basic proof methods in discre...

The human quest for discovering mathematical beauty in the arts

In the words of the twentieth-century British mathematician G. H. Hardy,...

On Radically Expanding the Landscape of Potential Applications for Automated Proof Methods

In this paper we examine the potential of computer-assisted proof method...

Please sign up or login with your details

Forgot password? Click here to reset