Recently, i've been thinking of the diophantine equation:
$A^{B^C}+D^{E^F} = G^{H^I}$
Where, $A, B, C, D, E, F, G, H, I \in [2, \infty)\cap \mathbb{N}$.
I discovered that one can construct a solution if one finds perfect powers $P, Q$ and a positive integer $n$ that satisfies $nP +1 = Q$. It is not hard to see that a solution to the diophantine equation follows from such $P, Q, n$.
$$(2^n)^P + (2^n)^P=2^Q$$
I call these the trivial solutions. My question is, are there any non-trivial solution to this diophantine equation?