Elliptic curves

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

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.

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

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

Mathematicians are working to fully explain unusual behaviors uncovered using artificial intelligence.

Landmark results in Ramsey theory and a remarkably simple aperiodic tile capped a year of mathematical delight and discovery.

Some strange mathematical sequences are always whole numbers — until they’re not. The puzzling patterns have revealed ties to graph theory and prime numbers, awing mathematicians.

Modular forms are one of the most beautiful and mysterious objects in mathematics. What are they?

Ana Caraiani and James Newton have extended an important result in number theory to the imaginary realm.