I'm working on a problem from a past exam paper,
Define the group operation on the set of solutions of Pell's equation, $$G_d=\{(x,y)\in\mathbb{Z}^2:x^2-dy^2=1\}$$ and show that the group axioms are satisfied.
I don't have access to course notes yet. I've surmised from this paper (adapted from their (1.2)) that the group operation required is $$(x_1,y_1)\star(x_2,y_2)=(x_1x_2+dy_1y_2,x_1y_2+x_2y_1).$$ Proving G1 and G2 is tedious but straightforward. I figured out G3. I so far haven't managed to figure out G4.
Best Answer
It's better to understand where $\star$ is from. Define $$\rho(x,y):=x + y \sqrt{d}$$ We have $$\rho((x_1, y_1)\star(x_2, y_2))=\rho(x_1, y_1)\rho(x_2, y_2).$$ Now everything follows naturally, since $\{x+y\sqrt d\,|\,(x,y)\in G_d\}$ forms a group under the usual multiplication of real numbers.
In particular, the inverse of $(x,y)$ under $\star$ is $(x, -y)$, as the inverse of $x+\sqrt dy$ under multiplication is $x-\sqrt dy$.