Given the equation $$ (2^n-1)(1+x+x^2+...x^{2k})(1+y+y^2+...+y^{2j})=2^nx^{2k}y^{2j}-1 $$ a non negative integer solution exists of the form $(2^n,2^{2kn+n}$). How can I (try to) show that (1) this is the only non negative solution and/or (2) that no non negative solutions exist with ${both} (x,y) $ odd parity?
My only attempt is to show that if a different solution existed it say $(p,q)$ where both are odd natural numbers that if $p$ is the smaller of the two then $p>2^n$ so $q<2^{2k+1}$ and somehow arrive at a contradiction but it seems tricky.