Let $(u,v)\in\mathbb{Z}^2\setminus(0,0)$ and $p$ a prime number congruent to 1 modulo 4. Show that if $p$ does not divide $u^2+v^2$, then $$|L_{\mathrm{prim}}(u,v,p)| \leq 2 \quad \text{and} \quad |L_{\mathrm{prim}}(u,v,p^2)| \leq 1.$$
Let $(u,v)\in\mathbb{Z}^2\setminus(0,0)$ and $p$ a prime number congruent to 1 modulo 4. Show that if $p$ does not divide $u^2+v^2$, then
$$|L_{\mathrm{prim}}(u,v,p)| \leq 2 \quad \text{and} \quad |L_{\mathrm{prim}}(u,v,p^2)| \leq 1.$$