Conjectures
Lonely Runner Conjecture
LOHN-lee RUN-er kon-JEK-cher
Also Known As View-Obstruction Problem
Combinatorics
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The lonely runner conjecture asks readers to imagine n runners, each running at a distinct constant speed around a circular track of unit length starting together at the same point. It asserts that every runner is, at some moment, at least 1/n of the track's length away from every other runner. The problem was first posed in purely number-theoretic terms in 1967 by Jorg Wills, and independently reformulated in 1974 as a view-obstruction problem by Thomas Cusick; the popular running-track formulation dates to 1998. It connects to Diophantine approximation, the study of how well real numbers can be approximated by fractions. The conjecture is proven for up to 13 runners as of this atlas's research pass, with several of those cases settled only in 2025 and 2026, but remains open for an arbitrary number of runners.
Facts
StatementGiven n runners starting at the same point and time on a circular track of unit length, each running at a distinct constant speed, every runner is, at some moment, at distance at least 1/n along the track from every other runner. 1 Proposed Year Prize StatusNot a Millennium Prize Problem; no institutional cash prize is attached to its proof. 1 Progress Toward ResolutionProven for up to 13 runners as of this atlas's research pass: n equal to 8 and 9 were proved in 2025 by Rosenfeld, and by Rosenfeld and Trakulthongchai independently, with n equal to 10 through 13 following in 2025 and 2026. Terence Tao proved improved asymptotic lower bounds in 2018, and Malikiosis, Santos and Schymura reduced the number of cases that would need checking to confirm the conjecture in general to n raised to the power 2n, in 2025. 1 Reported by the atlas as of 2026; the small-n proofs are recent (2025 to 2026) and the frontier may move again. Cross-Tradition Connections
In Branch
Posed By
Posed the conjecture in 1967 in purely number-theoretic terms.
Independently formulated the equivalent view-obstruction problem in 1974.
Sources
1. Lonely Runner Conjecture (Wikipedia)
WikipediaFormulation sectionQuote, Formulation section
n runners on a track of unit length, with constant speeds all distinct from one another, will each be lonely at some time, at least 1/n units away from all others
1. Lonely Runner Conjecture (Wikipedia)
WikipediaHistory sectionQuote, History section
The conjecture was first posed in 1967 by German mathematician Jorg Wills, in purely number-theoretic terms, and independently as a view-obstruction problem in 1974 by Thomas W. Cusick.
1. Lonely Runner Conjecture (Wikipedia)
Wikipedialead sectionQuote, lead section
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 n runners on a track of unit length, with constant speeds all distinct from one another, will each be lonely at some time, at least 1/n units away from all others.
1. Lonely Runner Conjecture (Wikipedia)
WikipediaFor specific n section, and Tighter bounds sectionQuote, For specific n section, and Tighter bounds section
The conjecture is true for n <= 13 runners.
1. Lonely Runner Conjecture (Wikipedia)
Wikipedialead section and History sectionQuote, lead section and History section
Its illustrative and now-popular formulation dates to 1998, though the conjecture originated in 1967 as part of work in Diophantine approximation.
1. Lonely Runner Conjecture (Wikipedia)
WikipediaOpening paragraphQuote, Opening paragraph
independently as a view-obstruction problem in 1974 by Thomas W. Cusick
1. Lonely Runner Conjecture (Wikipedia)
Open Questions (1 open question)
Does every runner on a circular track, moving at its own constant and distinct speed, really become lonely at some moment, for every number of runners n?
The conjecture is proved individually for up to 13 runners, most recently n = 10 through 13 in 2025 and 2026, and Tao proved an improved general asymptotic bound in 2018, but no argument covers every n at once.
What would resolve this A proof, or a counterexample, covering every number of runners n rather than a further specific value.
CombinatoricsLonely Runner Conjecture (Wikipedia)
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