Israeli scientists have developed algorithms for finding mathematical formulas in the form of infinite fractions, which use such fundamental constants as π or e. The approach was named the Ramanujan Machine in honor of the brilliant Indian mathematician Srinivasa Ramanujan, who was able to intuitively obtain complex and most often correct mathematical expressions, not proving them. The researchers' article was published in the journal Nature.
As the authors write, the algorithms are able to find dozens of well-known formulas, as well as previously unknown in the form of continued fractions, which are representations of π, e, Catalan constant (the sum of an infinite alternating series) and values of zeta functions. Some of the generated mathematical hypotheses have already been proven, while others remain neither proven nor disproved.
In the search for hypotheses, a combination of two algorithms was used: a variation of the meet-in-the-middle algorithm and a gradient descent-type optimization algorithm adapted to the recurrent structure of continued fractions. Both work on the basis of enumerating numerical values, therefore they generate formulas without proof and do not require prior knowledge of the so-called mathematical structure (that is, how the relationship between the elements of an expression should be built). However, this methodology can be used in conjunction with automated theorem proving.
The Ramanujan Machine itself is implemented in the form of distributed computing, when volunteers donate the computing resources of personal computers to search for new expressions. You can join the community on the project website. Participants can also propose proofs of derived formulas or new algorithms.
The authors point out that the Ramanujan Machine is changing the approach in formal proofs where sequential logic is applied. Algorithms use numerical data to derive mathematical structures, mimicking the intuition of great mathematicians, and allow further mathematical research.