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A science fiction novelist and an internet commenter made breakthroughs on a longstanding problem about the number of ways you can arrange a set of items. What did they discover?

Several mathematicians under the age of 30 left their marks all over the field, and amateur problem-solvers of all ages made significant contributions to long-dormant puzzles.

Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work.

Equations, like numbers, cannot always be split into simpler elements.

Tadashi Tokieda discovers new physical phenomena by looking at the everyday world with the eyes of a child.

Through exacting geometric calculations, Philip Gibbs has found the smallest known cover for any possible shape.

Alexander Smith’s work on the Goldfeld conjecture reveals fundamental characteristics of elliptic curves.

A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years.

A new statistical model appears to undermine long-held assumptions in number theory. How much should it be trusted when all that really matters is proof?