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.