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Setting Limits on Supersymmetry Using Simplified Models
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Generating conjectures on fundamental constants with the Ramanujan Machine.

Gal Raayoni1, Shahar Gottlieb1, Yahel Manor1,2

  • 1Technion-Israel Institute of Technology, Haifa, Israel.

Nature
|February 4, 2021
PubMed
Summary
This summary is machine-generated.

Algorithms can now discover new mathematical formulas for fundamental constants like pi and e. This systematic approach, called the Ramanujan Machine, uncovers hidden structures and previously unknown equations.

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Area of Science:

  • * Mathematics and computational science, with applications across physics, biology, and chemistry.

Background:

  • * Discoveries of new mathematical formulas relating fundamental constants (e.g., pi, e) have historically been rare and sporadic.
  • * Such discoveries often relied on mathematical ingenuity or profound intuition, rather than systematic methods.

Purpose of the Study:

  • * To propose a systematic, algorithm-driven approach for discovering mathematical formulas for fundamental constants.
  • * To reveal underlying mathematical structures and complement traditional proof-based methodologies.

Main Methods:

  • * Development and application of the 'Ramanujan Machine,' utilizing algorithms to identify novel formulas.
  • * Implementation of a meet-in-the-middle algorithm variant and a tailored gradient descent optimization algorithm.
  • * Algorithms are based on numerical value matching, enabling conjecture generation without prior structural knowledge.

Main Results:

  • * Discovery of numerous well-known and previously unknown formulas, including continued fraction representations for pi, e, Catalan's constant, and Riemann zeta function values.
  • * Generation of mathematical conjectures, some readily provable and others remaining open problems.
  • * Demonstration of the algorithms' effectiveness in uncovering structures for constants with unknown mathematical underpinnings.

Conclusions:

  • * The Ramanujan Machine offers a systematic method for mathematical formula discovery, augmenting human intuition.
  • * This algorithmic approach reverses conventional logic in proofs by using numerical data to unveil structures.
  • * The methodology provides new avenues for mathematical research, particularly for constants with unknown properties.