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Generating conjectures on fundamental constants with the Ramanujan Machine

机译：用苎麻师机器产生基本常数的猜想

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摘要

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~(3-6). 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 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 proving~(8-13). 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.
机译：e和π等基本数学常数在不同的科学领域中，从抽象数学和几何到物理，生物学和化学〜（1,2）。然而，对于几个世纪以来，新的数学公式相关的基本常数稀缺，通常偶尔发现〜（3-6）。这种发现通常被认为是高斯和ramanujan〜（7）如高斯和ramanujan〜（7）的大量数学家的数学聪明语或深刻的直觉的行为。在这里，我们提出了一种系统的方法，可以利用算法来发现基本常数的数学公式，并有助于揭示常量的底层结构。我们称之为“ramanujan机器”。我们的算法发现了几十个众所周知的公式以及先前未知的公式，例如π，e，加泰罗尼亚州的持续的持续分数表示和riemann zeta函数的值。我们的算法发现了几个猜想（回想起来）易于证明，而其他人则仍然是未经制造的。我们提出了两种证明，该算法在寻找猜想中有用：相遇中算法的变种和梯度下降优化算法，对持续分数的反复化结构定制。这两种算法都基于匹配的数值;因此，它们猜测公式而不提供证据或需要先前了解潜在的数学结构，使得该方法互补地与自动定理证明〜（8-13）。当申请发现没有已知数学结构的基本常数时，我们的方法是特别有吸引力的，因为它逆转了正式证据中顺序逻辑的传统用途。相反，我们的工作支持不同的研究概念框架：计算机算法使用数值数据来揭示数学结构，从而试图替换大量数学家的数学直觉，并提供导致进一步的数学研究。

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  • 来源
    《Nature》 |2021年第7844期|67-73|共7页
  • 作者

    Gal Raayoni; Shahar Gottlieb; Yahel Manor; George Pisha; Yoav Harris; Uri Mendlovic; Doron Haviv; Yaron Hadad; Ido Kaminer;

  • 作者单位

    Technion-Israel Institute of Technology;

    Technion-Israel Institute of Technology;

    Technion-Israel Institute of Technology|The Technion Harry and Lou Stern Family Science and Technology Youth Center Pre-University Education;

    Technion-Israel Institute of Technology;

    Technion-Israel Institute of Technology;

    Google;

    Technion-Israel Institute of Technology;

    Technion-Israel Institute of Technology;

    Technion-Israel Institute of Technology;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);美国《生物学医学文摘》(MEDLINE);美国《化学文摘》(CA);
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