Number theory

Using high school algebra and geometry, and knowing just one rational point on a circle or elliptic curve, we can locate infinitely many others.

Laurent Fargues and Peter Scholze have found a new, more powerful way of connecting number theory and geometry as part of the sweeping Langlands program.

Jordan Ellenberg enjoys studying — and writing about — the mathematics underlying everyday phenomena.

Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.

Despite finding no specific examples, researchers have proved the existence of a pervasive kind of prime number so delicate that changing any of its infinite digits renders it composite.

Enter the world of perfect numbers and explore the mystery mathematicians have spent thousands of years trying to solve.

A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.

A group of MIT undergraduates is searching for tetrahedra that tile space, the latest effort in a millennia-long inquiry. They’ve already made a new discovery.

Four mathematicians have cataloged all the tetrahedra with rational angles, resolving a question about basic geometric shapes using techniques from number theory.