Algorithm-assisted discovery of an intrinsic order among mathematical constants

Abstract

In recent decades, a growing number of discoveries in mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces. As computers become more powerful, an intriguing possibility arises—the interplay between human intuition and computer algorithms can lead to discoveries of mathematical structures that would otherwise remain elusive. Here, we demonstrate computer-assisted discovery of a previously unknown mathematical structure, the conservative matrix field. In the spirit of the Ramanujan Machine project, we developed a massively parallel computer algorithm that found a large number of formulas, in the form of continued fractions, for numerous mathematical constants. The patterns arising from those formulas enabled the construction of the first conservative matrix fields and revealed their overarching properties. Conservative matrix fields unveil unexpected relations between different mathematical constants, such as π and ln(2), or e and the Gompertz constant. The importance of these matrix fields is further realized by their ability to connect formulas that do not have any apparent relation, thus unifying hundreds of existing formulas and generating infinitely many new formulas. We exemplify these implications on values of the Riemann zeta function ζ (n), studied for centuries across mathematics and physics. Matrix fields also enable new mathematical proofs of irrationality. For example, we use them to generalize the celebrated proof by Apéry of the irrationality of ζ (3). Utilizing thousands of personal computers worldwide, our research strategy demonstrates the power of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.