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

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 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.

Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work.

Equations, like numbers, cannot always be split into simpler elements.

Quanta’s In Theory video series returns with an exploration of a mysterious mathematical pattern found throughout nature.

When a crystallographer treated prime numbers as a system of particles, the resulting diffraction pattern created a new view of existing conjectures in number theory.

A simple, step-by-step breakdown of two “perfect” math proofs.

Generations of researchers have pursued his “Langlands program,” which seeks to create a grand unified theory of mathematics.