Primes of the form $x^2+ny^2+mz^2$ and congruences.
Mostly, you should look at a number of items at
(address updated in 2018):
http://zakuski.math.utsa.edu/~kap/
including Dickson_Diagonal_1939.pdf and Kap_Jagy_Schiemann_1997.pdf to begin with.
Now that I think of it, you also need to read Kap_All_Odd_1995.pdf at the same place, also a new preprint by Jeremy Rouse on his 451 Theorem(s), as it is readily decided whether a form represents the single number 2. In a different direction, you need Wai Kiu Chan and Byeong-Kweon Oh, "Positive Ternary Quadratic Forms with Finitely Many Exceptions," Proceedings of the A.M.S., Volume 132, Number 6, Pages 1567-1573 (2004).
The overriding fact is that ternary forms, like binary forms, are collected together in genera. Unlike binary forms, these vary in size (class number) for a fixed discriminant. The good thing is the result of Jones, every number given by congruence conditions (a finite number of "progressions") is, in fact, represented by at least one form in the genus.
So, an example, $x^2 + 4 y^2 + 9 z^2$ is not regular, so it is not in Dickson's list. The genus of this form represents all numbers not of shape $9 n \pm 3, \; 8 n + 3, \; 4^k (8n+7).$ The other class in the genus is $x^2 + y^2 + 36 z^2.$ Between the two forms, all eligible numbers are represented. It is not difficult to prove that $x^2 + 4 y^2 + 9 z^2$ misses only the single number 2 out of the eligible numbers. So, if you were so minded, you could say that $x^2 + 4 y^2 + 9 z^2$ represents all primes that pass the above restrictions as well as not being $0 \pmod 2.$ As the restriction $9n \pm 3$ does not affect larger primes, just larger composite numbers, one could also say that $x^2 + 4 y^2 + 9 z^2$ represents all primes $p \equiv 1 \pmod 4.$
So there is a built in problem with your formulation. If one of your positive ternaries $x^2 + m y^2 + n z^2$ has finitely many exceptions, in particular finitely many prime exceptions $p_1,p_2,\ldots,p_k,$ the form can be said to represent every prime $p$ fitting the original restrictions and the new restrictions $p \neq 0 \pmod {p_k.}$ As a result, your list of ternary forms is infinite and unprovable. As Franz says, you need a tighter formulation.
If you like, email me your list of forms, we can discuss it.
A better example of the possible horror: Ono and Soundararajan (1997) showed that Ramanujan's form $x^2 + y^2 + 10 z^2$ has only squarefree numbers as sporadics (numbers represented by some form in the genus but not by this form, "exceptions" for Chan and Oh). They also showed that GRH implies that the known list is complete. So, GRH implies that $x^2 + y^2 + 10 z^2$ represents all primes not divisible by any of 3, 7, 31, 43, 67, 79, 223, 307, 2719. The other sporadics are composite. At the same time, it is easy to show that the form represents all numbers $n \equiv 5 \pmod 6,$ first pointed out in a letter from J.S.Hsia to Kaplansky, later a cheap proof by me, and one by Oh.
EDIT: it occurs to me that an alternate property could be: a positive ternary form will be defined to be fungible if its sporadics are all composite. Or perhaps funicular. I looked it up, the best would be frangible.
EDIT TOOOOO: I thought I might find examples of forms $x^2 + m y^2 + n z^2$ that seem to be fungible, or perhaps funicular, or frangible, despite lacking proof. The first example is $$ x^2 + y^2 + 48 z^2 \neq 21 \cdot 9^k $$ compared with the other form in that genus, $2 x^2 + 2 y^2 + 13 z^2 + 2 y z + 2 z x,$ checked on numbers up to 1,250,000. Very similar, $$ x^2 + 4y^2 + 20 z^2 \neq 77 $$ compared with the other form in that genus, $4 x^2 + 4y^2 + 5 z^2,$ also checked on numbers up to 1,250,000. In this second case it is easy to show that each form of the genus represents 4 times any number represented by the other form, and no numbers $2 \pmod 4$ are represented anyway, so only odd numbers come up. Anyway, 21 and 77 are composite. I have not proved these completely, just checked on computer.
EDIT TOOTOOTOO: I got an opinion from Jeremy Rouse. He points out that any positive ternary has two possible causes for having infinitely many numbers missed (compared to its genus), those being high divisibility by anisotropic primes or spinor exceptional classes. These two phenomena affect only finitely many squareclasses. In both cases, we do not increase the set of primes missed, with the result that a positive ternary fails to represent only a finite number of eligible (by congruence conditions) primes. This also explains, to some degree, the reference to Duke and Schulze-Pillot (1990). The final corollary says that any sufficiently large number that is primitively represented by some form in the same spinor genus is represented by the form of interest. There are only a few spinor exceptional squareclasses, so, even in an irregular spinor genus, we can only miss finitely many squarefree numbers, as those other than the spinor exceptional integers are represented by something in the same spinor genus, and primes are squarefree and therefore represented primitively if at all. I think I've caught up now. Note the D_S-P results give no effective bound, so we cannot identify the primes missed without some fortunate accident such as regularity, spinor regularity, regularity with regard to all odd numbers, and so forth.
Dear Joel,
I noticed your request for texts. The most informative chapter on positive ternaries, with the intent of predicting the represented integers, is in Dickson M.E.N.T. (1939). I have typed up a list of my books. For the moment, my websites are down, the host computer fried a power supply. So, I am including the link to my preprints on the arXiv, the papers with Alex Berkovich may be just the thing, overlap of modular forms and quadratic forms. The fundamental result is the weighted representation measure of Siegel. Again, as far as numbers integrally represented, the books of Jones, Watson, and Cassels are most helpful. I'm also including the Lattice website, although the emphasis there is classifying interesting lattices (positive forms) rather than finding the numbers represented (squared norms, often just called norms). I've included SPLAG and Ebeling, again I do not mainly use the lattice viewpoint, but there you go.
Carl Ludwig Siegel
Lectures on the Analytical Theory of Quadratic Forms (Second Term 1934/35)
Leonard Eugene Dickson
Studies in the Theory of Numbers (1930)
Modern Elementary Theory of Numbers (1939)
Burton Wadsworth Jones
The Arithmetic Theory of Quadratic Forms (1950)
George Leo Watson
Integral Quadratic Forms (1960)
John William Scott Cassels
Rational Quadratic Forms (1978)
Jean-Pierre Serre
A Course in Arithmetic (English translation 1973)
John Horton Conway
The Sensual Quadratic Form (1997)
Sphere Packings, Lattices and Groups (1988, with Neil J.A. Sloane)
Wolfgang Ebeling
Lattices and Codes (2nd, 2002)
Gordon L. Nipp
Quaternary Quadratic Forms (1991)
O. Timothy O'Meara
Introduction to Quadratic Forms (1963)
Yoshiyuki Kitaoka
Arithmetic of Quadratic Forms (1993)
Larry J. Gerstein
Basic Quadratic Forms (2008)
http://arxiv.org/find/math/1/au:+Jagy_W/0/1/0/all/0/1
http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/
There are also some excellent, influential books by Lam. I sometimes ask him questions, the response is typically that he does not do forms over rings. So, among many threads that might be called quadratic forms, I put Lam in with the name Pfister. Again, I recently got involved with the lattice viewpoint, see SPLAG and Ebeling. The trick there is that it is possible to relate ideas such as covering radius to class number. This relationship is so easy that there really ought to be a short article on "here is how you do this, which you would never know by surveying the literature." But the entire matter is dismissed in a single paragraph on page 378 of SPLAG. When asking for help, I told Richard Borcherds that I sometimes wanted to write a book Here's how YOU can do quadratic forms, and he agreed that one can do a good deal with very little machinery.
Yes. See "Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids", Duke and Schulze-Pillot.