Assume that there is a positive integer solution, then pick one that has to beminimum. We have that since otherwise we would be able to cancel by a common factor, contradicting that is mimimal. Now write . This is a Pythagorean equation, there are two cases either is even or odd. If odd then for positive integers and . But then . Thus, we found another solution to this Diophantine equation which is even smaller then the original, thus we have a contradiction by infinite descent. Now you do the case when is even.