Determine all integers $x,y$ such that $f:=(3^x-y^3)(x^3-3^y)$ is a perfect square.
What I have thought: the problem seems a bit hard to begin with. I first tried the situation that $\gcd(x,3)=\gcd(y,3)=1$ and was stuck. It would help a lot if we can make some progress in $\gcd(3^x-y^3,x^3-3^y)$, but I failed. However, if you just expand it, it still seems useless.
Any idea which makes progress is welcomed.