Number theory

A straightforward conjecture about runners moving around a track turns out to be equivalent to many complex mathematical questions. Three new proofs mark the first significant progress on the problem in decades.

Mathematicians are still trying to understand fundamental properties of the Fourier transform, one of their most ubiquitous and powerful tools. A new result marks an exciting advance toward that goal.

Explore a shape that can’t pass through itself, a teenage prodigy, and two new kinds of infinity.

Using a relatively young theory, a team of mathematicians has started to answer questions whose roots lie at the very beginning of mathematics.

The proof, known to be so hard that a mathematician once offered 10 martinis to whoever could figure it out, connects quantum mechanics to infinitely intricate mathematical structures.

By extending the scope of the key insight behind Fermat’s Last Theorem, four mathematicians have made great strides toward building a “grand unified theory” of math.

A new proof illuminates the hidden patterns that emerge when addition becomes impossible.

By proving a broader version of Hilbert’s famous 10th problem, two groups of mathematicians have expanded the realm of mathematical unknowability.

It’s surprisingly difficult to prove one of the most basic properties of a number: whether it can be written as a fraction. A broad new method can help settle this ancient question.