This came up in proving non-regularity of a certain language (powers of 2 over the ternary alphabet). Any clue to the above equation could help me move forward.

Edit:
Of course, $x = 1, y = 1$ is a solution. I am looking for non-trivial solutions.

Except $x=1$ and $y=1$ there aren't any.

We have that

$$3^{2x} - 1 = 8^y$$

i.e

$$ (3^x + 1)(3^x - 1)  = 8^y$$

Thus we must have that

$$3^x + 1 = 2^m, 3^x - 1 = 2^n$$

Thus $$2^m - 2^n = 2$$

i.e
$$ 2^n(2^{m-n} - 1) = 2$$

Thus $n=1$ and $m=2$.
