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.