I don't have an algebraic solution to this problem. If you try squaring both sides to get rid of square roots, you'll end up with an equation involving . So I think you'll probably have to resort to a bit of intelligent trial and error.
If this equation is going to have any exact integer solutions then will have to be a perfect square. The only candidates are , giving possible values of as
In fact and are solutions, and I'm certain they are the only ones.
I cheated a bit and used a spreadsheet to have a look at the function
When and for and is increasing more slowly than . So I'm sure that there are no further solutions.
I don't know whether anyone else can come up with a more analytical solution.