Ko [Ko40] proved there are none if $(x,y)=1$, but there are in fact infinitely many solutions in general - for example, \[x=2^{12}3^6, y = 2^83^8,\textrm{ and } z = 2^{11}3^7.\] More generally, writing $a=2^{n+1}$ and $b=2^n-1$, \[x = 2^{a(b-n)}b^{2b}\cdot 2^{2n},\] \[y = 2^{a(b-n)}b^{2b}\cdot b^2,\] and \[z = 2^{a(b-n)}b^{2b}\cdot 2^{n+1}b.\] In [Er79] Erdős asks if the infinite families found by Ko are the only solutions.