Solve for positive real x
There may be a quick proof using fixed-point theorems, but let’s consider the following algebraic method.
Since the LHS is an integer, all solutions are (positive) integers. I claim that all solutions are of the form
Note that This is because for all
And because
i.e.
This proves that all are solutions. Now we must show that there are no other solutions.
Any other solution would have to be of the form where Suppose there is such a solution.
Again we have since for all
Then, in order for the solution to hold, we would need to have This would mean that we would need
i.e.
i.e.
i.e.
Now and
If then
If then
In either case, we have a clear contradiction of
This completes the proof.