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.


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    $\begingroup$ 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. $\endgroup$ – Franz Lemmermeyer Oct 11 '13 at 10:34
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    $\begingroup$ 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 $b^2 - 4ac$. $\endgroup$ – Franz Lemmermeyer Oct 11 '13 at 10:34
  • $\begingroup$ @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. $\endgroup$ – Pete L. Clark Oct 11 '13 at 15:29
  • $\begingroup$ 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... $\endgroup$ – Pete L. Clark Oct 11 '13 at 15:32
  • $\begingroup$ @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. $\endgroup$ – Will Jagy Oct 11 '13 at 18:12

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.

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    $\begingroup$ Thanks, Franz. Pete will be writing to you, he has several questions remaining. Something about indefinite binary forms either achieving their supremum or not, for which your 1995 Exposition. Math. paper is the main reference. Meanwhile, I was able to download Weber 1882 from the Gottingen digital library, also Briggs 1954 from something Canada. $\endgroup$ – Will Jagy Oct 12 '13 at 19:31

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