Combinatorics

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.”

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.

A new proof breaks a decades-long drought of progress on the problem of estimating the size of triangles created by cramming points into a square.

The numbers count a variety of seemingly unrelated mathematical structures.

A new result shows how much of the plane can be colored by points that are never exactly one unit apart.

Mathematicians can often figure out what happens as quantities grow infinitely large. What about when they are just a little big?

A new paper finds a faster method for determining when two mathematical groups are the same.

Mathematicians probe the limits of randomness in new work estimating quantities called Ramsey numbers.