Number theory

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.

Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future.

To make progress on one of number theory’s most elementary questions, two mathematicians turned to an unlikely source.

Two mathematicians have renewed a debate about the fundamental nature of some of math’s most important equations.

Born poor in colonial India and dead at 32, Ramanujan had fantastical, out-of-nowhere visions that continue to shape the field today.

A new proof about prime numbers illuminates the subtle relationship between addition and multiplication — and raises hopes for progress on the famous abc conjecture.

In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.

The proof creates stricter limits on potential exceptions to the famous Riemann hypothesis.