What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be "none". The generalization by Ramanujan only seems to cover non-negative coefficients.



Four-Square Theorem for Negative Coefficient
With indefinite forms, it is possible for ternary forms to be universal. Indeed, all are known. References are given in *Modern Elementary Theory of Numbers* by Leonard Eugene Dickson, (1939). With any integer $M$ and any **odd** $N,$ they are equivalent to (by an invertible linear change of variables) one of four:


$$ xy - M z^2, $$


$$ 2xy - N z^2, $$


$$ 2 xy + y^2 - N z^2, $$


$$ 2 xy + y^2 - 2 N z^2. $$


If you allow $M=0$ you get a universal binary.


There are universal quaternary forms without such a universal ternary "section," for example $$ w^2 - 2 x^2 + 3 y^2 - 6 z^2. $$ 


EEEEEDDDDIITTTTT:


$2xy+ z^2$ is integrally equivalent to $a^2 + b^2 - c^2,$ by


$$ 
 \left( \begin{array}{ccc}
 1 & 0 & 1 \\
 0 & -1 & 1 \\
 1 & -1 & 1 
\end{array} 
 \right) 
 \left( \begin{array}{ccc}
 0 & 1 & 0 \\
 1 & 0 & 0 \\
 0 & 0 & 1 
\end{array} 
 \right) 
 \left( \begin{array}{ccc}
 1 & 0 & 1 \\
 0 & -1 & -1 \\
 1 & 1 & 1 
\end{array} 
 \right) 
 =
 \left( \begin{array}{ccc}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & -1 
\end{array} 
 \right). 
 $$


