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.