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Let $Q$ be an indefinite binary quadratic form with discriminant $D$ and one class per genus (keep the example $x^2 - 2y^2$ in mind). If one asks about the set $P = \{ p : p \text{ prime and } p = Q(x,y) \text{ for some } x,y \in \mathbb{Z} \}$. It's pretty easy to determine the relative density of $P$ in the set of all primes. However, suppose that we look at the set $P'(B) = \{ p : p \text{ prime and } p = Q(x,y) \text{ for some } x,y \in \mathbb{Z}, |x|,|y| \le B \}$ of primes represented by $Q$ where $(x,y)$ is in a box. Is it still true that $\lim_{B \rightarrow \infty} |P'(B)|/\pi(C(B)) = \lim_{N \rightarrow \infty} |\{ p \in P, |p| \le N \}|/\pi(N)$, where $C(B) = \max_{|x|,|y| \le B} |Q(x,y)|$?

I'm not completely sure of what should be the right denominator in the first limit. Perhaps one should look at $P'(B,N) = \{ p : p \text{ prime and } p = Q(x,y) \text{ for some } x,y \in \mathbb{Z}, |x|,|y| \le B, p \le N\}$ and have $B$ be a function of $N$, and ask for which functions the two limits are equal.

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It's going to take me a while, but this is one case where Conway's topograph method may do the trick. Marty Weissman is partway through writing a book elaborating on that method and others. Of course, he is currently wandering around Singapore as well. Anyway, what you get to do is investigate over a single cycle, that is $PSL(2,\mathbb Z)$ mod the automorphish group of the form. – Will Jagy Jul 8 '13 at 18:39
I appreciate the shout out, and I agree that looking modulo the orthogonal group of the form is a good idea. The topograph gives a good way of seeing a fundamental domain for this orthogonal group acting on primitive pairs $(x,y)$ but I'd need to run some experiments to make a guess on heuristics of $P'(B)$. – Marty Jul 9 '13 at 0:41

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