Number theory

It’s been known for thousands of years that the primes go on forever, but new proofs give fresh insights into how theorems depend on one another.

Using “refreshingly old” tools, mathematicians resolved a 50-year-old conjecture about how to categorize important functions called modular forms, with consequences for number theory and theoretical physics.

Mathematicians are taking ideas developed to study random numbers and applying them to a broad range of categories.

Four Fields Medals were awarded for major breakthroughs in geometry, combinatorics, statistical physics and number theory, even as mathematicians continued to wrestle with how computers are changing the discipline.

The work — the first-ever limit on how many whole numbers can be written as the sum of two cubed fractions — makes significant headway on “a recurring embarrassment for number theorists.”

In his senior year of high school, Daniel Larsen proved a key theorem about Carmichael numbers — strange entities that mimic the primes.

Two mathematicians have proven Patterson’s conjecture, which was designed to explain a strange pattern in sums involving prime numbers.

Ever since Archimedes, mathematicians have been fascinated by equations that involve a difference between squares. Now two mathematicians have proven how often these equations have solutions, concluding a decades-old quest.

The Kakeya conjecture predicts how much room you need to point a line in every direction. In one number system after another — with one important exception — mathematicians have been proving it true.