Need a proof to show all the units are satisfied $\mathbf{Z}\sqrt{2}$ is the all the integer solution in Pell equation

Question

We know the integer solutions of Pell's equation $$a^2-2b^2=\pm1$$ correspond to the units of $\textbf{Z}[\sqrt{2}]$. How can we prove this?

Answer 1

Probably you mean if $(a,b)$ is a solution of Pell's equation then $a+b\sqrt{2}$ is a unit of $\mathbb{Z}\sqrt{2}$. This directly follows from the fact$$(a+b\sqrt{2})(a-b\sqrt{2})=\pm 1$$