Lonely runner conjecture

Unsolved problem in mathematics
Is the lonely runner conjecture true for every number of runners?
<a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">More unsolved problems in mathematics</a>

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, specifically the study of <a href="/facts/Diophantine_approximation/WGVMuWF4">Diophantine approximation</a>, the lonely runner conjecture is a <a href="/facts/Conjecture/oPWUb7SF">conjecture</a> 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 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 seven 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 <a href="/facts/Chromatic_number/J6HoBfP0">chromatic numbers</a>, of certain graphs.

Lonely runner conjecture open-in-new

Lonely runner conjecture