Number theory

Two mathematicians have renewed a debate about the fundamental nature of some of math’s most important equations.

Born poor in colonial India and dead at 32, Ramanujan had fantastical, out-of-nowhere visions that continue to shape the field today.

A new proof about prime numbers illuminates the subtle relationship between addition and multiplication — and raises hopes for progress on the famous abc conjecture.

In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.

The proof creates stricter limits on potential exceptions to the famous Riemann hypothesis.

In 1940 André Weil wrote a letter to his sister, Simone, outlining his vision for translating between three distinct areas of mathematics. Eighty years later, it still animates many of the most exciting developments in the field.

Mathematicians have illuminated what sets of points can look like if the distances between them are all whole numbers.

Mathematicians are working to fully explain unusual behaviors uncovered using artificial intelligence.

Simple rules in simple settings continue to puzzle mathematicians, even as they devise intricate tools to analyze them.