can you generalize the method of solving the equation of the type $2^x$ + $19^y$ = $z^2$ has a solution (3, 0, 3). Also, $8^x$ + $19^y$ = $z^2$ has no solution. But, $8^x$ + $17^y$ = $z^2$ has a solution (2, 1, 9). I think these there are very close to each other in means of prime base of each y term.
What is the important role by Catalan conjecture in this questions. Instead of trail and error can we write some general solutions of those equations. Can you discuss...plz?