Exercise 2.9.7

Proof. Write $\sqrt{\mathrm{\Delta }}$ a root of $\mathrm{\Delta }$ in $ℂ$.

If $(x,y)\in {\mathbb{Z}}^2$ and $(x^{\prime },y^{\prime })\in {\mathbb{Z}}^2$ are solutions of the Pell equation $x^2-y^2\mathrm{\Delta }=\pm 4$, then

Then

Therefore

It remains to show that $\left(\frac{x{x}^{\prime }+\mathrm{\Delta }y{y}^{\prime }}{2},\frac{x{y}^{\prime }+x^{\prime }y}{2}\right)\in {\mathbb{Z}}^2$.

$\text{∙}$

If $\mathrm{\Delta }\equiv 0(\mathit{mod}4)$, then $x^2-\mathrm{\Delta }{y}^2=\pm 4$ implies $x^2\equiv 0(\mathit{mod}4)$, thus $x$ is even, and similarly $x^{\prime }$ is even. This shows that $x{x}^{\prime }+\mathrm{\Delta }y{y}^{\prime }$ and $x{y}^{\prime }+x^{\prime }y$ are even.

$\text{∙}$

If $\mathrm{\Delta }\equiv 1(\mathit{mod}4)$, then $x^2-\mathrm{\Delta }{y}^2=\pm 4$ implies $x^2-y^2\equiv 0(\mathit{mod}4)$, thus $x,y$ have the same parity, and similarly $x^{\prime },y^{\prime }$ have the same parity, and $$x{x}^{\prime }+y{y}^{\prime }\mathrm{\Delta }\equiv 2x{x}^{\prime }\equiv 0(\mathit{mod}2),x{y}^{\prime }+x^{\prime }y\equiv 2\mathit{xy}\equiv 0(mod2).$$

In both cases, $\left(\frac{x{x}^{\prime }+\mathrm{\Delta }y{y}^{\prime }}{2},\frac{x{y}^{\prime }+x^{\prime }y}{2}\right)\in {\mathbb{Z}}^2$.

To conclude, $\left(\frac{x{x}^{\prime }+\mathrm{\Delta }y{y}^{\prime }}{2},\frac{x{y}^{\prime }+x^{\prime }y}{2}\right)$ is also a solution of $x^2-\mathrm{\Delta }{y}^2=\pm 4$.

Recall that the matrix $U=U(f,x,y)$ is given by (2.20):

Then

where

This proves $U(f,x,y)U(f,x^{\prime },y^{\prime })=U(f,(x{x}^{\prime }+y{y}^{\prime }\mathrm{\Delta })∕2,(x{y}^{\prime }+x^{\prime }y)∕2)$. □