Ramanujan Machine

The Ramanujan Machine, published in Nature on February 4, 2021, is a conceptual framework named after the Indian mathematician Srinivas Ramanujan, which generates conjectures of continued fraction representations of mathematical constants. Based on several search and optimization algorithms the Ramanujan Machine generates conjectured representations of well known fundamental mathematical constants.

Theory
In mathematics, a conjecture is a proposition that is suspected to be true due to preliminary supporting evidence, and it forms a major step in developing new theories in order to prove them. Many conjectures throughout history led to significant mathematical advances as these required the development of new tools. Trying to formalise the process of conjecture generation the researchers built a number of optimization and search algorithms to find new representations of important constants such as π, Euler's number e, Catalan’s constant or values of the Riemann zeta function. Although the Ramanujan Machine does not prove the conjectures it can be used in conjunction with Automated theorem proving (ATP) algorithms to generate an End-to-End framework of generating and proving mathematical equations and identities.

Concisely, the algorithm can be described as follows. For a given mathematical constant $$c$$ (such as π or e) the search algorithm aims to find an expression following:

$$ \frac{\gamma(c)}{\delta(c)} = \text{PCF}(\alpha, \beta) $$

where $$\alpha, \beta, \gamma, \delta$$ are integer polynomials and $$\text{PCF}(\alpha, \beta)$$ is the continued fraction:

$$ \text{PCF}(\alpha, \beta) = a_0 + \frac{b_1}{a_1 + \frac{b_2}{a_2 + \frac{b_3}{a_3 + \ldots}}} $$

with $$a_n = \alpha(n)$$ and $$b_n = \beta(n)$$. The algorithm then consists of trying many different polynomials $$\alpha, \beta, \gamma, \delta$$ until a continued fraction representation for the constant $$c$$ is found.

Development
The Ramanujan Machine was developed by a team of researchers at the Israel Institute of Technology.

Criticism
Frank Calegari published two criticisms of the Raayoni et al. preprint that first described the Ramanujan Machine. Frank Calegari's criticism focuses on the approach the authors of the paper took toward promoting their work. He states that "the choice to deliberately obscure the fact that the program has generated (as yet) nothing considered remotely new by an expert while simultaneously boasting of a triumph in automating intuition is not just absurd, but is an intellectual fraud. Ramanujan would roll over in his grave."

Part of Calegari's criticism stems from the fact that the algorithm merely suggests special cases of Gauss continued fraction, which has been known since 1813. A later version of the Raayoni et al. paper, published February 2021, presents new results addressing these remarks. This criticism was also discussed during the Nature peer-review process.

Contrary to the criticism by Frank Calegari, George Andrews (mathematician) is quoted saying, "The fact that they have improved the irrationality exponent for the Catalan constant from 0.554 to 0.567 reveals that they are able to make contributions to really hard problems."