Number theory

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.

Terence Tao, who has been called the “Mozart of Mathematics,” wrote an essay in 2007 about the common ingredients in “good” mathematical research. In this episode, the Fields Medalist joins Steven Strogatz to revisit the topic.

Landmark results in Ramsey theory and a remarkably simple aperiodic tile capped a year of mathematical delight and discovery.

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.