Show that $\varepsilon=(3+\sqrt{7}) /(3-\sqrt{7})$ is a unit in $k=\mathbb{Q}(\sqrt{7})$. Show further that 2 is the square of the principal ideal in $k$ generated by $3+\sqrt{7}$.

Assuming that the Minkowski constant for $k$ is $\frac{1}{2}$, deduce that $k$ has class number 1 .

Assuming further that $\varepsilon$ is the fundamental unit in $k$, show that the complete solution in integers $x, y$ of the equation $x^{2}-7 y^{2}=2$ is given by

$$x+\sqrt{7} y=\pm \varepsilon^{n}(3+\sqrt{7}) \quad(n=0, \pm 1, \pm 2, \ldots) .$$

Calculate the particular solution in positive integers $x, y$ when $n=1$