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.


[I WILL FIND OUT](http://www.youtube.com/watch?v=Uhh3nNcstkw)



primes represented by an indefinite binary quadratic form
Meyer (Über einen Satz von Dirichlet, Crelle 103 (1888)) proved that a primitive
binary quadratic form with nonsquare discriminant represents infinitely many primes that
lie in any given compatible residue class modulo a given integer N. He did so only for forms with even middle coefficient, and did not address the signs of primes, but both of these require only minor modifications. 


This answer makes me wonder whether someone actually proved, with Dirichlet's methods, that a
finite number of forms with coprime discriminants simultaneously represent infinitely many primes - by Chebotarev, this has to be true. 


