About representations of $s$ and $-s$ of indefinite quadratic forms

Question

Given any indefinite binary integral quadratic form $f(x,y)=ax^2+bxy+cy^2$ with discriminant $D=b^2-4ac$ and $a,b,c$ not all zero. Does there always exists a $s \in \mathbb{Z}$ such that $s$ is represented by $f$ but $-s$ is not. $s$ is allowed to depend on the quadratic form $f$. I am sorry for the bad title, however, I don't know how to title the question.

Answer 1

what happens with $$ x^2 - 5 y^2 $$

suggest writing a little program, let $x$ go from $0$ up to $50,$ then for each $x$ let $y$ increase from $0$ until stopping when $x^2 - 5 y^2 < -100.$ Along the way, record any values of $n = x^2 - 5 y^2$ with $-100 \leq n \leq 100.$ Then print out those numbers in order

Answer 2

Some quadratic forms $f$ are multiplicative; that is, for any numbers $m$ and $n$ represented by $f$, $f$ represents $mn$, too. This answer shows how to prove that one particular quadratic form is multiplicative. It can be adapted for others. Now if $f$ is multiplicative and represents $-1$...