Number theory

Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades.

We finally know how big a set of numbers can get before it has to contain a pattern known as a “polynomial progression.”

Looking for answers in infinite space is hard. High school math can help narrow your search.

The twin primes conjecture is one of the most important and difficult questions in mathematics. Two mathematicians have solved a parallel version of the problem for small number systems.

The way you learned to multiply works, but computers employ a faster algorithm.

The ancient Greeks wondered when “irrational” numbers can be approximated by fractions. By proving the longstanding Duffin-Schaeffer conjecture, two mathematicians have provided a complete answer.

Decades after the landmark proof of Fermat’s Last Theorem, ideas abound for how to make it even more reliable. But such efforts reflect a deep misunderstanding of what makes the proof so important.

It’s an educated guess, not a proof. But a good conjecture will guide math forward, pointing the way into the mathematical unknown.

By chopping up large numbers into smaller ones, researchers have rewritten a fundamental mathematical speed limit.