### Lonely Runner Conjecture- II

#### by ayushkhaitan3437

The Lonely Runner conjecture states that each runner is lonely at some point in time. Let the speeds of the runners be , and let us prove “loneliness” for the runner with speed .

As we know, the distance between and is given by the formula for and for , where stands for time. Also, .

therefore when . Also, therefore . Finally, observing that this is a periodic process with a period of , we come upon the conclusion that is lonely as compared to when , where .

Now we prove the loneliess of with respect to every other runner. This is equivalent to the statement

, where . Also, note that all can take different integral values.

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