Number theory

Ana Caraiani seeks to unify mathematics through her work on the ambitious Langlands program.

New work establishes a tighter connection between the rank of a polynomial and the extent to which it favors particular outputs.

Legend says the Chinese military once used a mathematical ruse to conceal its troop numbers. The technique relates to many deep areas of modern math research.

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.