The approaches that you are using are called Elementary solutions to diophantine equations. They are the best ones but often the most difficult methods. According to E.T. Bell, Euler has spend 7 years solving this equation with elementary techinques. This shows how difficult it is without advanced methods in number theory.
But I was not asking for an elementary approach, for I belive it is too long. I was asking for other approaches I have never seen before.
