I was wondering if there exists a proof showing that there exists no Pythagorean Triple such that when its bases are swapped with the exponents the left and right hand side of the Pythagorean Triple are still equal to one another.
In other words, is there a proof showing $$2^x + 2^y = 2^z$$ for a Pythagorean Triple $$x^2 + y^2 = z^ 2?$$ I have searched everywhere for such a proof but I can't seem to find one and I didn't want to start writing my own proof in case there already exists one.
Hint: The only way to satisfy $2^x+2^y=2^z$ with positive integers is for $x=y$.