Suppose I have a form $$ f(x,y) = a x^2 + b x y + c y^2, $$ with $a,b,c$ integers, $\gcd(a,b,c)=1$ and $\Delta = b^2 - 4 a c > 0,$ but $\Delta \neq n^2$ for any integer $n.$

Do there exist (positive) primes $p,q$ such that $f$ integrally represents $p$ and $-q?$

I have most books on quadratic forms of which I've ever heard, but I do not see this. I will keep checking. For positive forms we have Chebotarev density and infinitely many primes. And, of course, this may also follow from Chebotarev density; if so, is there a cheaper way as well?

I would like to allow odd $b,$ but I suppose it does not really matter: $f$ represents a superset of the numbers represented by $f(x,2y), f(2x,y), f(x-y,x+y),$ one of which is primitive.

reference request: in principle I know how to prove the result. But I assume it is well-known and don't want to spend the two pages or so that it would take to include a complete proof. $\endgroup$3more comments