“To get effective results on rational points, it definitely has the feeling that there’d have to be a new idea,” said Ellenberg.

At present, there are two main proposals for what that new idea could be. One comes from the Japanese mathematician Shinichi Mochizuki, who in 2012 posted hundreds of pages of elaborate, novel mathematics to his faculty webpage at Kyoto University. Five years later, that work remains largely inscrutable. The other new idea comes from Kim, who has tried to think about rational numbers in an expanded numerical setting where hidden patterns between them start to come into view.

A Symmetry Solution

Mathematicians often say that the more symmetric an object is, the easier it is to study. Given that, they’d like to situate the study of Diophantine equations in a setting with more symmetry than the one where the problem naturally occurs. If they could do that, they could harness the newly relevant symmetries to track down the rational points they’re looking for.

To see how symmetry helps a mathematician navigate a problem, picture a circle. Maybe your objective is to identify all the points on that circle. Symmetry is a great aid because it creates a map that lets you navigate from points you do know to points you have yet to discover.

Imagine you’ve found all the rational points on the southern half of the circle. Because the circle has reflectional symmetry, you can flip those points over the equator (changing the signs of all the y coordinates), and suddenly you’ve got all the points in the northern half too. In fact, a circle has such rich symmetry that knowing the location of even one single point, combined with knowledge of the circle’s symmetries, is all you need to find all the points on the circle: Just apply the circle’s infinite rotational symmetries to the original point.

Yet if the geometric object you’re working with is highly irregular, like a random wandering path, you’re going to have to work hard to identify each point individually — there are no symmetry relationships that allow you to map known points to unknown points.

Sets of numbers can have symmetry, too, and the more symmetry a set has, the easier it is to understand — you can apply symmetry relationships to discover unknown values. Numbers that have particular kinds of symmetry relationships form a “group,” and mathematicians can use the properties of a group to understand all the numbers it contains.

The set of rational solutions to an equation doesn’t have any symmetry and doesn’t form a group, which leaves mathematicians with the impossible task of trying to discover the solutions one at a time.

Beginning in the 1940s, mathematicians began to explore ways of situating Diophantine equations in settings with more symmetry. The mathematician Claude Chabauty discovered that inside a larger geometric space he constructed (using an expanded universe of numbers called the p-adic numbers), the rational numbers form their own symmetric subspace. He then took this subspace and combined it with the graph of a Diophantine equation. The points where the two intersect reveal rational solutions to the equation.

In the 1980s the mathematician Robert Coleman refined Chabauty’s work. For a couple of decades after that, the Coleman-Chabauty approach was the best tool mathematicians had for finding rational solutions to Diophantine equations. It only works, though, when the graph of the equation is in a particular proportion to the size of the larger space. When the proportion is off, it becomes hard to spot the exact points where the curve of the equation intersects the rational numbers.

“If you have a curve inside an ambient space and there are too many rational points, then the rational points kind of cluster and you have trouble distinguishing which ones are on the curve,” said Kiran Kedlaya, a mathematician at the University of California, San Diego.

And that’s where Kim came in. To extend Chabauty’s work, he wanted to find an even larger space in which to think about Diophantine equations — a space where the rational points are more spread out, allowing him to study intersection points for many more kinds of Diophantine equations.