Solutions existence of quadratic diophantine equations

What can be said on existence of at least one integer solution of $$N = Ax^2 + By^2 + Cz^2$$ where $N, A, B, C$ are given positive integer numbers?

In other words, is there any criteria whether integer $N$ can be represented in such form, when $A, B, C$ are given.

It could also be helpful if either

• $N, A, B, C$ are generally integer, not only positive, or
• there are only two variables, i.e. equation is $N = Ax^2 + By^2$

I have seen this question: Existence of solutions to diophantine quadratic form but I can't see exactly how it can help me.

There are 102 forms $0 < A \leq B \leq C, \gcd(A,B,C) = 1$ where your first question has a definitive answer. Probably worth saying this: still with $0 < A \leq B \leq C, \gcd(A,B,C) = 1$ we can give a definitive answer to $$A x^2 + B y^2 + C z^2 = N w^2.$$ This is very similar to the question linked in your question. We cannot usually say whether it is possible to demand $w=1.$
Let me add this: when two of $A,B,C$ are positive and one negative, but we still have $\gcd(A,B,C) = 1 .$ As long as $ABC$ is not divisible by $128$ or by any $p^3$ for odd prime $p,$ we can tell exactly what $N$ can be represented using congruences.
• @MikhailKuzin That is about right, given a random postivie ternary we probably cannot say what numbers it represnts. Given a random indefinite rernary, we probably can say what numbers it representes, but that is a good deal of work. there are also a few irregular (positive) ternary forms for which we can prove what numbers are represented, for example $x^2 + 4 y^2 + 9 z^2.$ The situation is very, very different for indefinite forms. In case of interest, I put a number of article pdf's at zakuski.math.utsa.edu/~kap Jun 21 '17 at 20:42
• @MikhailKuzin I should emphasize this type of example: $x^2 + 3 y^2 - 3 z^2$ is definitely not universal, it is never $2 \pmod 3.$ However, we can easily tell what numbers are represented. Try some experiments on computer, what numbers between $-1000$ and $1000$ are represented with $x,y,z < 100?$ Jun 21 '17 at 21:02