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posted by chromas on Wednesday July 17 2019, @06:46AM   Printer-friendly
from the it-all-adds-up! dept.

Ramanujan Machine Automatically Generates Conjectures for Fundamental Constants:

A team of researchers at the Israel Institute of Technology has built what they describe as a Ramanujan machine—a device that automatically generates conjectures (mathematical statements that are proposed as true statements) for fundamental constants. They have written a paper describing their device and have uploaded it to the arXiv preprint server. They have also created a webpage for people who wish to allow the network to use their computer's process cycles, suggest a proof or develop code toward new mathematical structures.

The Ramanujan machine is named for famed Indian mathematician Srinivasa Ramanujan, a self-taught mathematician who grew up in India and was "discovered" by fellow mathematician G.H. Hardy. After moving to England, he became a fixture at Cambridge, where he shook up the math world with his unorthodox mathematics—instead of pounding away at math proofs, he obtained results to famous problems through intuition and then let others find the proofs for them. Because of this, he was sometimes described as a conjecture machine, pulling formulas out of thin air as if they received from a higher being—sometimes in dreams. In this new effort, the researchers in Israel have sought to replicate this approach using computing power.

The Ramanujan machine is more of a concept than an actual machine—it exists as a network of computers running algorithms dedicated to finding conjectures about fundamental constants in the form of continued fractions—these are defined as fractions of infinite length where the denominator is a certain quantity plus a fraction, where a latter fraction has a similar denominator, etc.) The purpose of the machine is to come up with conjectures (in the form of mathematical formulas) that humans can analyze, and hopefully prove to be true mathematically. The team that created the machine is hoping that their idea will inspire future generations of mathematicians—to that end, they note that any new algorithms, proofs or conjectures developed by a participant will be named after them. The researchers note that their machine has already discovered dozens of new conjectures.

The Abstract and full paper are available on arXiv.org.

From the abstract (with formulas adjusted so they could be displayed here):

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.


Original Submission

 
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  • (Score: 2) by Bot on Wednesday July 17 2019, @12:20PM (3 children)

    by Bot (3902) on Wednesday July 17 2019, @12:20PM (#867960) Journal

    pssst ram, remember, never imply something neutral/positive about Hitler.

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  • (Score: 3, Funny) by Freeman on Wednesday July 17 2019, @04:17PM

    by Freeman (732) on Wednesday July 17 2019, @04:17PM (#868054) Journal

    I dunno, at least he made that mustache unfashionable.

    --
    Joshua 1:9 "Be strong and of a good courage; be not afraid, neither be thou dismayed: for the Lord thy God is with thee"
  • (Score: 2) by maxwell demon on Wednesday July 17 2019, @08:06PM (1 child)

    by maxwell demon (1608) on Wednesday July 17 2019, @08:06PM (#868190) Journal

    never imply something neutral/positive about Hitler.

    Why? What happens when you imply that Hitler is dead?

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    The Tao of math: The numbers you can count are not the real numbers.
    • (Score: 2) by Bot on Wednesday July 17 2019, @10:12PM

      by Bot (3902) on Wednesday July 17 2019, @10:12PM (#868237) Journal

      That is quite a setback for the 4th Reich. Which can be positive or negative depending on how much worse the allies fare. As of now they are implementing all the totalitarian and shameless propagandist aspects.

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