Say we have as ranges over positive reals, where is a fixed positive real.

Say that is irrational. In this case we can set to be such that , . Ranging over all such , we have comes arbitrarily close to so we can have while and . No irrational solutions.

Say that is rational. Then we have and we can turn into .

So then assume we had for all positive where WLOG . Then just set and , and ranging we have that keeps incrementing by of the circle. So we can increase it by multiples of freely and then if we can pick which gives us a contradiction. So we have that or , and also or . Now, then is , , , or . Note the is just the and the and cases are the same by symmetry.

It suffices to examine and .

Just pick , and done with the first one.

So it suffices to check if the second one is true. Note it is actually just the case.

We wonder if holds for all positive reals . If we plug in though, we have is positive, but whereas , so oops. So no solutions exist.