Solution

in other words, it is a pells equation.

The fundamental solution is $(x_1,y_1) = (8,3)$, and then all solutions are given by $(x_{n+1},y_{n+1}) = (8x_n + 21y_n,3x_n+8y_n)$.

The norm in $\mathbb{Z}[\sqrt{7}]$ is multiplicative, so since $N(8+3\sqrt{7})=1$, we have $N((8+3\sqrt{7})^n)=1$ for all positive integers $n$.