Mathematics

Just as ice melts to water, graphs undergo phase transitions. Two mathematicians showed that they can pinpoint such transitions by examining only local structure.

The Markov numbers reveal the secrets of irrational numbers and the patterns of the Fibonacci sequence. But there’s one question about them that has resisted proof for over a century.

A team of four prominent mathematicians, including two Fields medalists, proved a conjecture described as a “holy grail of additive combinatorics.”

Researchers prove that navigating certain systems of vectors is among the most complex computational problems.

Mathematicians prove a theorem that illuminates the geometry of universes with tiny amounts of mass.

How Fermat’s less famous "little theorem" got mathematicians young and old to play with prime-like Carmichael numbers.

Some strange mathematical sequences are always whole numbers — until they’re not. The puzzling patterns have revealed ties to graph theory and prime numbers, awing mathematicians.

High-dimensional spheres can have a much wider variety of structures than mathematicians thought possible.

Quadratic reciprocity lurks around many corners in mathematics. By proving it, number theorists reimagined their whole field.