Lonely runner conjecture
number-theoretic conjecture that states that 𝑘 people running around a circular track with distinct speeds will each be, at some point, separated by ¹⁄ₖ from every other runner
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
{\displaystyle 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
{\displaystyle 1/n}
units away from all others.
The conjecture was first posed in 1967 by German mathematician Jörg Wills, in purely number-theoretic terms, and independently as a view-obstruction problem in 1974 by Thomas W. Cusick; its illustrative and now-popular formulation dates to 1998. The conjecture is known to be true for
13
{\displaystyle 13}
runners or fewer, but the general case remains unsolved. Implications of the conjecture include solutions to view-obstruction problems and bounds on properties, related to chromatic numbers, of certain graphs.
Formulation
Consider
n
{\displaystyle n}
runners on a circular track of unit length. At the initial time
t
=
0
{\displaystyle t=0}
, all runners are at the same position and start to run; the runners' speeds are constant, all distinct, and may be negative. A runner is said to be lonely at time
t
{\displaystyle t}
if they are at a distance (measured along the circle) of at least
1
/
n
{\displaystyle 1/n}
from every other runner. The lonely runner conjecture states that each runner is lonely at some time, no matter the choice of speeds.
This visual formulation of the conjecture was first published in 1998.
“Lonely runner conjecture” enters the record as number-theoretic conjecture that states that 𝑘 people running around a circular track with distinct speeds will each be, at some point, separated by ¹⁄ₖ from every other runner. Crown Archives preserves that source wording while asking what Lonely, runner and conjecture can confirm, complicate or overturn.
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The citation trail is more important than the brevity of the summary: it shows where individual claims can be examined in context. The source revision retrieved here is dated Aug 28, 2026. The linked authority identifier is Q6671695. None of the 1 selected statements returned an explicit reference. The first chronological checks are 1967, 1974 and 1998.
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