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 chemistry^{1,2}. Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and usually discovered sporadically^{3,4,5,6}. Such discoveries are often considered an act of mathematical ingenuity or profound intuition by great mathematicians such as Gauss and Ramanujan^{7}. 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 meetinthemiddle 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 proving^{8,9,10,11,12,13}. 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.
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All the results of the Ramanujan Machine project are shared in the paper, with newer updates appearing periodically on the project website.
Code availability
Code is available at: http://www.ramanujanmachine.com/ and the GitHub links therein.
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Acknowledgements
We thank M. Soljačić, B. Weiss, D. Soudry and D. Carmon for helpful discussions. I.K. is grateful for the support of R. Magid and B. Magid and for the support of the Azrieli Faculty Fellowship. Y.M. acknowledges the support and guidance of the Israeli Alpha Program for Excellent HighSchool Students.
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G.R., G.P. and I.K. implemented the first proofofconcept algorithms. G.R. implemented the first generation MITMRF algorithm. S.G. and Y. Harris implemented the stateoftheart MITMRF algorithm. S.G. made the developments that led to the discovery of the ζ(3) and Catalan PCFs. Y.M. implemented the Descent&Repel algorithm. Y.M., S.G., U.M. and I.K found how to convert the Catalan PCFs into expressions with record approximation exponents and fast convergence rates. U.M., Y.M., G.R., S.G., Y. Harris and I.K. proposed parts of the algorithms and developed proofs for some of the conjectures. D.H. and Y. Hadad developed the online community. Y. Hadad, G.P. and I.K. came up with the conceptual flow of the wider concept. I.K. conceived the idea and led the research. All authors provided substantial input to all aspects of the project and to the writing of the paper.
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Peer review information Nature thanks YangHui He, Doron Zeilberger and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
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Extended data figures and tables
Extended Data Fig. 1 Convergence rates of the PCFs.
The plots present the absolute difference between the PCF value and the corresponding fundamental constant (that is, the error) versus the number of terms calculated in the PCF. On the left are PCFs with exponential/superexponential convergence rates, and on the right are PCFs that converge polynomially. The majority of previously known PCFs for π converge polynomially, whereas all of our newly found results converge exponentially.
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Supplementary Information
This file contains Supplementary Sections A–G, 6 Supplementary Tables, and a Supplementary Figure. The first section provides the results found by the Ramanujan Machine algorithms. The last section provides the new findings about the Catalan constant.
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Raayoni, G., Gottlieb, S., Manor, Y. et al. Generating conjectures on fundamental constants with the Ramanujan Machine. Nature 590, 67–73 (2021). https://doi.org/10.1038/s41586021032294
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DOI: https://doi.org/10.1038/s41586021032294
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