The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants
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Fundamental mathematical constants like e and π are ubiquitous in diverse fields of science, from abstract mathematics and geometry to physics, biology and chemistry. Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and usually discovered sporadically. In this paper we propose a novel and systematic approach that leverages algorithms for deriving new mathematical formulas for fundamental constants and help reveal their underlying structure. Our algorithms find dozens of well-known as well as previously unknown continued fraction representations of π, e, and the Riemann zeta function values. Two new conjectures produced by our algorithm, along with many others, are: e = 3 + -1/4+-2/5+-3/6+-4/7+... , 4/π-2 = 3+1·3/5+2· 4/7+3· 5/9+4· 6/11+... We present two algorithms that proved useful in finding new results: a variant of the Meet-In-The-Middle (MITM) algorithm and a Gradient Descent (GD) tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values and thus find new conjecture formulas without providing proofs and without requiring prior knowledge on any mathematical structure. This approach is especially attractive for fundamental constants for which no mathematical structure is known, as it reverses the conventional approach of sequential logic in formal proofs. Instead, our work presents a new conceptual approach for research: computer algorithms utilizing numerical data to unveil new internal structures and conjectures, thus playing the role of mathematical intuition of great mathematicians of the past, providing leads to new mathematical research.
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