I have seen $3^x$ + $3^y$ = $6^z$ and $4^x$ + $18^y$ = $22^z$ on lecture series of Prof. Gandhi. In my own study, I have constructed the following theorem (I am not sure about solvability) and I am seeking some good discussion on this theorem including proof/ comments etc.
If A = {(2k-1, 2k-1, $2^k$)|k is a positive integer}, B = {(2k+3, 2k, 3.$2^k$)|k is a non negative integer} and C = {2k, 2k+3, 3.$2^k$)|k is a non negative integer}, then the solution of $2^x$ + $2^y$ = $z^2$ is (x, y, z) belongs to $A \cup B \cup C$.
Also discuss that, if P is prime and > 2 then solution set of $2^x$ + $p^y$ = $z^2$ will be...
Thanks in advance.