(a) Let $N$ be a non-square integer. Describe the integer solutions of the Pell equation $x^{2}-N y^{2}=1$ in terms of the convergents to $\sqrt{N}$. Show that the set of integer solutions forms an abelian group. Denote the addition law in this group by $\circ$; given solutions $\left(x_{0}, y_{0}\right)$ and $\left(x_{1}, y_{1}\right)$, write down an explicit formula for $\left(x_{0}, y_{0}\right) \circ\left(x_{1}, y_{1}\right)$. If $(x, y)$ is a solution, write down an explicit formula for $(x, y) \circ(x, y) \circ(x, y)$ in the group law.

(b) Find the continued fraction expansion of $\sqrt{11}$. Find the smallest solution in integers $x, y>0$ of the Pell equation $x^{2}-11 y^{2}=1$. Use the formula in Part (a) to compute $(x, y) \circ(x, y) \circ(x, y)$.