The artificial intelligence (AI) program DeepMind has gotten closer to proving a math conjecture that’s bedeviled mathematicians for decades and revealed another new conjecture that may unravel how mathematicians understand knots.

The two pure math conjectures are the first-ever important advances in pure mathematics (or math not directly linked to any non-math application) generated by artificial intelligence, the researchers reported Dec. 1 in the journal Nature. Conjectures are mathematical ideas that are suspected to be true but have yet to be proven in all circumstances. Machine-learning algorithms have previously been used to generate such theoretical ideas in mathematics, but thus far these algorithms have tackled problems smaller than the ones DeepMind has cracked.

“What hasn’t happened before is using [machine learning] to make significant new discoveries in pure mathematics,” said Alex Davies, a machine-learning specialist at DeepMind and one of the authors of the new paper.

Much of pure mathematics is noticing patterns in numbers and then doing painstaking numerical work to prove whether those intuitive hunches represent real relationships. This can get quite complicated when working with elaborate equations in multiple dimensions.

However, “the kind of thing that machine learning is very good at, is spotting patterns,” Davies told Live Science.

The first challenge was setting DeepMind onto a useful path. Davies and his colleagues at DeepMind worked with mathematicians Geordie Williamson of the University of Sydney, Marc Lackenby of the University of Oxford, and András Juhász, also of the University of Oxford, to determine what problems AI might be useful for solving.

They focused on two fields: knot theory, which is the mathematical study of knots; and representation theory, which is a field that focuses on abstract algebraic structures, such as rings and lattices, and relates those abstract structures to linear algebraic equations, or the familiar equations with Xs, Ys, pluses and minuses that might be found in a high-school math class.

Knotty problems

In understanding knots, mathematicians rely on something called invariants, which are algebraic, geometric or numerical quantities that are the same. In this case, they looked at invariants that were the same in equivalent knots; equivalence can be defined in several ways, but knots can be considered equivalent if you can distort one into another without breaking the knot. Geometric invariants are essentially measurements of a knot’s overall shape, whereas algebraic invariants describe how the knots twist in and around each other.

“Up until now, there was no proven connection between those two things,” Davies said, referring to geometric and algebraic invariants. But mathematicians thought there might be some kind of relationship between the two, so the researchers decided to use DeepMind to find it.