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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. ==The conjecture== Consider ''k'' 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'' at time ''t'' if he is at a distance of at least 1/''k'' from every other runner at time ''t''. The lonely runner conjecture states that each runner is lonely at some time. A convenient reformulation of the problem is to assume that the runners have integer speeds,〔This reduction is proved, for example, in section 4 of the paper by Bohman, Holzman, Kleitman〕 not all divisible by the same prime; the runner to be lonely has zero speed. The conjecture then states that for any set ''D'' of ''k'' − 1 positive integers with gcd 1, : where ||''x''|| denotes the distance of real number ''x'' to the nearest integer. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lonely runner conjecture」の詳細全文を読む スポンサード リンク
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