Mathematicians Make New Progress on the Longstanding Lonely Runner Conjecture

Wed, 22 Apr 2026 02:28:00 +0200

The “lonely runner” problem is a deceptively simple mathematical puzzle that has challenged researchers for decades. It imagines several runners moving around a circular track at different constant speeds and asks whether each runner will eventually be far enough from all others—specifically, at least a distance of 1/N away, where N is the number of runners. Although easy to state, the problem connects deeply to multiple areas of mathematics, including number theory, geometry, and graph theory, and has proven extremely difficult to solve in general.

Mathematicians have gradually extended proofs for increasing numbers of runners over time. Early results handled small cases, and by 2007 the conjecture had been proven for up to seven runners. Progress then stalled for nearly two decades. Recently, however, significant breakthroughs have revived interest in the problem. In 2025, mathematician Matthieu Rosenfeld proved the conjecture for eight runners using a combination of number theory a