I have been trying to prove that has no solutions in naturals numbers with
I first thought of finding a mod in which it makes no sense, but I just couldn't find one (after some attempts).
Thanks in advance
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.