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  • $\begingroup$ What is $h(-N)$? $\endgroup$ Commented Jul 19, 2017 at 17:08
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    $\begingroup$ If we look at prime discriminants of size between $X$ and $2X$, there are about $X/\log X$ such discriminants, and the class numbers are of size $\sqrt{X}$ typically. So a random model would suggest that the chance of finding primes $N$ of size $X$ is about $1/\log X \times 1/\sqrt{X} \times 1/\sqrt{X} = 1/(X\log X)$. In other words, we might expect to find a prime of size $X$ about $1/\log X$ times. Summing this over $X=2^k$ gives a barely divergent series, and so ``Borel-Cantelli" suggests that there are infinitely many such $N$. Of course this is delicate, but we don't know any better. $\endgroup$ Commented Jul 19, 2017 at 17:34
  • $\begingroup$ The comment above suggests that there might be about $\log \log X$ primes $N$ below $X$ meeting the conditions of the question. $\endgroup$ Commented Jul 19, 2017 at 17:37
  • $\begingroup$ @YCor If N is a square-free integer other than 1, then h(N) is the class number of the quadratic field Q(root(N)). In the case at hand, h(-N) is the number of SL_2(Z)-classes of positive integral binary quadratic forms of discriminant 4N, and h(-2N) the number of classes of such forms of discriminant 8N. $\endgroup$ Commented Jul 19, 2017 at 19:58