Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In number theory, and especially the study of diophantine approximation, the lonely runner conjecture is a conjecture originally due to J. M. Wills in 1967. Applications of the conjecture are widespread in mathematics; they include view obstruction problems and calculating the chromatic number of distance graphs and circulant graphs. The conjecture was given its picturesque name by L. Goddyn in 1998.Consider k + 1 runners on a circular track of unit length. At t = 0, all runners are at the same position and start to run; the runners'' speeds are pairwise distinct. A runner is said to be lonely if she is at distance of at least 1/(k + 1) from each other runner. The lonely runner conjecture states that every runner gets lonely at some time.
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