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.
ABSTRACT
Fundamental mathematical constants such as e and π are ubiquitous in diverse fields of science, from abstract mathematics and geometry to physics, biology and chemistry1,2. Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and usually discovered sporadically3-6. Such discoveries are often considered an act of mathematical ingenuity or profound intuition by great mathematicians such as Gauss and Ramanujan7. Here we propose a systematic approach that leverages algorithms to discover mathematical formulas for fundamental constants and helps to reveal the underlying structure of the constants. We call this approach 'the Ramanujan Machine'. Our algorithms find dozens of well known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan's constant, and values of the Riemann zeta function. Several conjectures found by our algorithms were (in retrospect) simple to prove, whereas others remain as yet unproved. We present two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent optimization algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure, making this methodology complementary to automated theorem proving8-13. Our approach is especially attractive when applied to discover formulas for fundamental constants for which no mathematical structure is known, because it reverses the conventional usage of sequential logic in formal proofs. Instead, our work supports a different conceptual framework for research: computer algorithms use numerical data to unveil mathematical structures, thus trying to replace the mathematical intuition of great mathematicians and providing leads to further mathematical research.
