Researchers have built an artificial intelligence (AI) that can generate new mathematical formulae — including some as-yet unsolved problems that continue to challenge mathematicians.
The Ramanujan Machine is designed to generate new ways of calculating the digits of important mathematical constants, such as π or e, many of which are irrational, meaning they have an infinite number of non-repeating decimals.
The AI starts with well-known formulae to calculate the digits — the first few thousand digits of π, for example. From those, the algorithm tries to predict a new formula that does the same calculation just as well. The process produces a good guess called a conjecture — it is then up to human mathematicians to prove that the formula can correctly calculate the whole number.
The team began to make the conjectures public on the project’s website in 2019, and researchers have since proved several of them correct. But some remain open questions, including one on Apery’s constant, a number that has important applications in physics. “The last result, the most exciting one, no one knows how to prove,” says physicist Ido Kaminer, who leads the project at the Technion — Israel Institute of Technology in Haifa. The automated creation of conjectures could point mathematicians towards connections between branches of maths that people did not know existed, he adds.
The project — described1 in Nature on 3 February — is named after Srinivasa Ramanujan, an Indian mathematician who was active in the early twentieth century. Ramanujan rarely wrote the types of proof that appear in conventional maths papers. Instead, he filled entire notebooks with formulae that he believed came from a goddess who appeared in his dreams. His work has continued to inspire new research long after he died, aged 32, in 1920.
The techniques in the Ramanujan Machine’s algorithms existed before, says mathematician Doron Zeilberger at Rutgers University in Piscataway, New Jersey. “The novelty is to combine them in a unified framework.”
Continued fractions
The Ramanujan Machine currently has limited applications: so far, the algorithms can generate only formulae of a particular type, called continued fractions. These express a number as an infinite sequence of fractions nested in each other’s denominators.
Kaminer’s team has experimented with a range of algorithms for finding continued fractions, and applied them to various conceptually important numbers. One of them is Catalan’s constant, a number that originated from nineteenth-century Belgian mathematician Eugène Catalan’s studies.
Catalan’s constant is approximately 0.916, but it is so mysterious that no one has yet worked out whether it is rational — that is, whether it can be expressed as a fraction of two whole numbers. The best mathematicians have been able to do is prove that its ‘irrationality exponent’ — a measure of how hard it is to approximate a number using rationals — is at least 0.554. Proving that Catalan’s constant is irrational would be equivalent to proving that its irrationality exponent is greater than 1. Formulae generated by the Ramanujan Machine have enabled Kaminer’s team to improve slightly on the best human result, bringing the exponent up to 0.567.
“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,” says George Andrews, a mathematician who has helped to curate the posthumous publication of some of Ramanujan’s notebooks. But the contributions made so far are not of the calibre that using Ramanujan’s name would suggest, says Andrews. “Calling this the Ramanujan Machine is over the top,” says Andrews, who is at Pennsylvania State University in University Park.
Kaminer’s team plans to broaden the AI’s technique so that it can generate other kinds of mathematical formula.
Increasing complexity
Automated generation of conjectures is not the only place where computers are helping to advance maths. Although many mathematicians prefer working with pencil and paper, standard research practice in the field now includes the use of mathematical software that can, for example, manipulate complicated algebraic expressions.
Computer-aided calculations have played a crucial part in producing the proofs of several high-profile results. And more recently, some mathematicians have made progress towards AI that doesn’t just perform repetitive calculations, but develops its own proofs. Another growing area has been software that can go over a mathematical proof written by humans and check that it is correct.
“Eventually, humans will be obsolete,” says Zeilberger, who has pioneered automation in proofs and has helped confirm some of the Ramanujan Machine’s conjectures2. And as the complexity of AI-generated mathematics grows, mathematicians will lose track of what computers are doing and will be able to understand the calculations only in broad outline, he adds.
Andrews says that although computers might be able to come up with mathematical statements, and even prove that they are true, without human intervention, it is unclear whether they will be able to distinguish profound, interesting statements from merely technically correct ones. “Until I can detect a well-developed ‘sense of mathematical taste’ in AI, I expect its role to be that of an important auxiliary tool, not that of independent discoverer.”
