Given that no ternary quadratic form is universal and given the form $x^2 + axy - y^2$, how do I go about determining the integers that are not representable by that form?
Using the Conway-Schneeberger 15-Theorem, if the quadratic form represents all integers from the set $\{1,2,3,5,6,7,10,14,15\}$, it is univeral (i.e., represents all positive integers) and if there is only one integer $x$ in this set that is not representable then there is a quadratic form that represents all integers in that set other than $x$.
Questions:
- How do I apply this to determine integers not representable by $x^2 + axy - y^2$?
- Can this be addressed for general $a$? If not, can we solve this for specific $a$ such as, how do we determine integers not representable by $x^2 + 4xy - y^2$