The lonely runner conjecture
runners start together on a circular track of length 1 and run forever, each at its own constant speed, with all speeds different. Prove that for each runner there is a moment when every other runner is at distance at least along the track. The speeds show that cannot be replaced by anything larger.
History and references
Still open as of 2026.
Jörg Wills posed the problem in 1967 as a question in Diophantine approximation, and Thomas Cusick arrived at it independently a few years later through view obstruction problems. The picture of runners on a track dates from 1998. Classical proofs cover up to 7 runners, with the 7 runner case settled in 2008. Since 2025, computer-assisted proofs by Matthieu Rosenfeld and by Tanupat Trakulthongchai and coauthors, built on results that bound the size of a minimal counterexample, have extended the conjecture to 8 runners and, in recent preprints, to as many as 13. The general case is open.
References
Solutions 0
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