$D\in\Bbb N$ is an idoneal integer if $N\in\Bbb N$ can be written uniquely as $N=x^2\pm Dy^2$ then $N=2^mp^n$ where $p$ is odd prime and $n\geq0$ and $m\geq0$ holds.

Let the list of idoneal integers be $\mathcal D$. Let set of primes that can be represented by a $D\in\mathcal D$ be $\mathcal P_D$.

Is there an $N_0\in\Bbb N$ such that $\bigcup_{D\in\mathcal D}\mathcal P_D$ contain all the primes above $N_0$?

If not what classes of primes are missed out?