Geodesic
A note on the lonely runner conjecture
Journal of integer sequences, Tome 12 (2009) no. 4
Suppose $n$ runners having nonzero distinct constant speeds run laps on a unit-length circular track. The Lonely Runner Conjecture states that there is a time at which all the $n$ runners are simultaneously at least $1/(n+1)$ units from their common starting point. The conjecture has been already settled up to six ($n \le 6$) runners and it is open for seven or more runners. In this paper the conjecture has been proved for two or more runners provided the speed of the $(i+1)$th runner is more than double the speed of the $i$th runner for each $i$, arranged in increasing order.
Classification : 11B50, 11B75, 11A99
Keywords: integers, distance from the nearest integer 
@article{JIS_2009__12_4_a7,
     author = {Pandey,  Ram Krishna},
     title = {A note on the lonely runner conjecture},
     journal = {Journal of integer sequences},
     year = {2009},
     volume = {12},
     number = {4},
     zbl = {1233.11026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_4_a7/}
}
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Pandey,  Ram Krishna. A note on the lonely runner conjecture. Journal of integer sequences, Tome 12 (2009) no. 4. http://geodesic.mathdoc.fr/item/JIS_2009__12_4_a7/