Lonely runner conjecture: Difference between revisions

Revision as of 20:56, 22 May 2022

Revision as of 20:57, 22 May 2022

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In [[number theory]], specifically the study of [[Diophantine approximation]], the '''lonely runner conjecture''' is a [[conjecture]] about the long-term behavior of runners on a circular track. It states that <math>n</math> runners, with constant speeds all distinct from one another, will each be ''lonely'' at some time—at least <math>1/n</math> units away from all others.

The conjecture was first posed in 1967 by German mathematician Jörg M. Wills, in purely number-theoretic terms, and independently in 1974 by T. W. Cusick; its illustrative and now-popular formulation dates to 1998. The conjecture is known to be true for 7 runners or less, but remains unsolved in the general case. Implications of the conjecture include solutions to view obstruction problems and bounds on properties of certain graphs, related to [[chromatic number]]s, of certain graphs.

==Formulation==