Let $d$ be a squarefree integer. Suppose prime p can be written as $p = a^2 − db^2$ for some integers $a$ and $b$.
Determine the (lattice structure) of $A=\{(x, y)\in\Bbb Z^2 : p | x^2 − dy^2\}$ , when when $(d/p) = −1$
We have $(\frac{d}{p})(\frac{b^2}{p})=(\frac{a^2}{p})$ so $(d/p) = −1 \Rightarrow (\frac{b^2}{p}) = -(\frac{a^2}{p})=0$
But then $p^2|a^2$ and $p^2|b^2$ so $p|1$ a contradiction. So there is no such prime p. Does it mean that $S=\emptyset$?
Thank you for your help.