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.