Are we allowed to use Fermat's Last theorem?

and since .

Let for some integer .

Then,

Since the left-hand side is an odd number, is odd.

Let for some integer .

By expanding, we get that

Therefore, by dividing by four on both sides,

If , then which is impossible since .

Otherwise, and by Fermat's Last Theorem, there are no positive integer solutions to this equation.

P.S. I am sure that not having positive integer solutions can be proved using elementary means.