Number theory

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

Despite recent progress on the notorious Collatz conjecture, we still don’t know whether a number can escape its infinite loop.

For millennia, mathematicians have wondered whether odd perfect numbers exist, establishing an extraordinary list of restrictions for the hypothetical objects in the process. Insight on this question could come from studying the next best things.

A powerful technique called SAT solving could work on the notorious Collatz conjecture. But it’s a long shot.

Two mathematicians have proved the first leg of Paul Erdős’ all-time favorite problem about number patterns.

Why do mathematicians enjoy proving the same results in different ways?

In his rapid ascent to the top of his field, James Maynard has cut a path through simple-sounding questions about prime numbers that have stumped mathematicians for centuries.

Representation theory was initially dismissed. Today, it’s central to much of mathematics.

Vesselin Dimitrov’s proof of the Schinzel-Zassenhaus conjecture quantifies the way special values of polynomials push each other apart.