# primes represented by an indefinite binary quadratic form

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

• Isn't this just the theorem of Dirichlet, Schering, Weber and Meyer? It should be in Bachmann's "Theorie der quadratischen Formen", together with references. Oct 11, 2013 at 10:34
• Of course everything follows from Chebotarev if you look at the behaviour of primes in the Hilbert class field in the strict sense of the quadratic number field with discriminant $b^2 - 4ac$. Oct 11, 2013 at 10:34
• @Franz: Hi there. I will probably be writing to you soon to ask about something else about indefinite binary forms: Will and I are, at long last, putting finishing touches on our paper. For the thing that Will asked (which is really on my behalf, and is going in the paper), let me say that it is really a 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. Oct 11, 2013 at 15:29
• Continued: in the definite case, this came up in a small paper I wrote last year with some graduate student coauthors. Similar situation: this time I knew exactly how to prove it, but was happy to discover that it could easily and quickly be stitched together from results already in the literature: see Theorem 1 and Section 2.1 of math.uga.edu/~pete/GoNI_Submitv3.pdf. In fact I make references to two sources here; one of the sources works also with indefinite forms. So there are just a few references to Cox's book that we need to replace with something else... Oct 11, 2013 at 15:32
• @FranzLemmermeyer, thanks. I should have said that I have most books in English on this topic; the mentions of Weber make no explicit mention of indefinite forms, and I was not sure; also not sure about Schering and Meyer, so perhaps this result is not mentioned in anything I've read before. Oct 11, 2013 at 18:12