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Solutions to the simplest polynomial equations — called “roots of unity” — have an elegant structure that mathematicians still use to study some of math’s greatest open questions.

Legend says the Chinese military once used a mathematical ruse to conceal its troop numbers. The technique relates to many deep areas of modern math research.

By focusing on relationships between solutions to polynomial equations, rather than the exact solutions themselves, Évariste Galois changed the course of modern mathematics.

Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.

Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.

A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.

A group of MIT undergraduates is searching for tetrahedra that tile space, the latest effort in a millennia-long inquiry. They’ve already made a new discovery.

Four mathematicians have cataloged all the tetrahedra with rational angles, resolving a question about basic geometric shapes using techniques from number theory.

Long considered solved, David Hilbert’s question about seventh-degree polynomials is leading researchers to a new web of mathematical connections.

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