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$d\in\Bbb N$ is an idoneal integer if $N\in\Bbb N_{>1}$ 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 $65$ known idoneal integers be $d_1,\dots,d_{65}$. Let set of primes that can be represented by $d_i$ be $\mathcal P_i$.

Is there an $N_0\in\Bbb N$ such that $$\bigcup_{i\in\{1,\dots,65\}}\mathcal P_i$$ contain all the primes above $N_0$?

If not what classes of primes are missed out?

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    $\begingroup$ For each $d_i$ pick a prime factor $p_i$ and choose a quadratic non-residue class $\pmod {p_i}$ (if two of the $p_i$ are the same, use the same quadratic non-residue). Now use the Chinese Remainder Theorem and Dirichlet's theorem. $\endgroup$
    – Lucia
    Commented Dec 30, 2016 at 16:12

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