An equation to solve in natural numbers Some clues needed for the equation
$$(3+\sqrt{2})^x-(5-3\sqrt{2})^y=6+9\sqrt{2}, \quad x,y \in \mathbb{N}$$

Since you want to solve in natural numbers, notice that $0 < 5 - 3\sqrt{2} < 1$. This implies that $ 0 < ( 5 - 3 \sqrt{2} ) ^ y < 1$, and hence
$$6 + 9 \sqrt{2} < (3 + \sqrt{2} )^x < 7 + 9 \sqrt{2} $$
This forces $x = 2$, from which $y = 1$ (from Brian).
