Integers

By proving a broader version of Hilbert’s famous 10th problem, two groups of mathematicians have expanded the realm of mathematical unknowability.

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

Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.

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

Mathematicians thought they were on the cusp of proving a conjecture about the ancient structures known as Apollonian circles. But a summer project would lead to its downfall.

Ever since Archimedes, mathematicians have been fascinated by equations that involve a difference between squares. Now two mathematicians have proven how often these equations have solutions, concluding a decades-old quest.

A new proof significantly strengthens a decades-old result about the ubiquity of ways to represent whole numbers as sums of unit fractions.

Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

The solution to this month’s puzzle explains how to design a Sudoku square to figure out the likes and dislikes of four or eight finicky friends.