Arithmetic geometry is a vibrant field at the intersection of number theory and algebraic geometry, focussing on the study of polynomial equations and the distribution of their rational solutions.

One of the oldest and simplest problems in geometry has caught mathematicians off guard—and not for the first time. Since antiquity, artists and geometers have wondered how shapes can tile the entire ...

They made some progress, re-proving the conjecture in two dimensions using different techniques—ones they hoped would be applicable to the three-dimensional case. But then they hit a wall. “At some ...

The Collatz Conjecture is a deceptively simple math problem. It has only two rules. First, pick any number. If it's even, divide it by two. If it's odd, multiply it by three and add one. This will ...

Much of mathematics is driven by intuition, by a deep-rooted sense of what should be true. But sometimes instinct can lead a mathematician astray. Early evidence might not represent the bigger picture ...

This is a preview. Log in through your library . Abstract The present paper explores three interrelated topics in Popper's theory of science: (1) his view of conjecture, (2) the aim of science, and (3 ...

A mathematician has solved a 30-year-old problem at the boundary between mathematics and computer science. He used an innovative, elegant proof that has his colleagues marveling at its simplicity. Hao ...