$x^y-y^x=1$ for $x,y\in\Bbb Z$ and $x,y>1$ is clearly a special case of very well known Catalan's conjecture (now resolved). It seems to be very limited special case, but I was told by someone today that solution to this, appears simple, problem, is in fact equivalent to all of Catalan's conjecture. I had some hard time believing this, and I still doubt that this is true. When graphing solutions to this equation among real numbers it clearly looks like $y$ is bounded below 2 for all $x$. I tried to employ some analytical tools to show this last fact, but I didn't manage to get anywhere.
My question is, is this special case really equivalent to full Catalan's conjecture? If it's not, is there some elementary way of showing that it's only integer solution is $(3,2)$?
Thanks in advance.
