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.

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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.

  • $\begingroup$ Does it follow (along with comment from Dodsy) that for 102 regular forms mentioned above and for forms which satisfy condition in article we have criteria, but for any other form we generally can not say wheter there is a solution or not? $\endgroup$ Jun 21 '17 at 20:31
  • $\begingroup$ @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 $\endgroup$
    – Will Jagy
    Jun 21 '17 at 20:42
  • $\begingroup$ @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 $x,y,z < 100?$ $\endgroup$
    – Will Jagy
    Jun 21 '17 at 21:02

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