# Representation by a quadratic binary form

For $$m$$ a non-zero integer and discriminant $$d=b^2-4ac$$ congruent to either 0 or 1 modulo 4, show that m is properly represented by some binary quadratic form $$f(x)=ax^2+bxy+cy^2$$ if and only if the congruence $$z^2\equiv d\ (mod\ 4|m|)$$has a solution.

I have proved the forward half, but cannot prove the congruence implies the proper representation.

• What is $\;z\;$ , anyway? – DonAntonio May 12 '19 at 11:22
• z is can be any integer, just a place holder to demonstrate that d is a quadratic residue – W M Seath May 12 '19 at 11:23
• Oh...so write so! Write "...iff $\;d\;$ is a quadratic residue modulo $\;m\;$" . – DonAntonio May 12 '19 at 11:26

If $$z^2 = d + 4mt,$$ then $$z^2 - 4mt = d,$$ and the binary $$\langle m,z,t \rangle$$ or $$g(x,y) = m x^2 + z xy + t y^2$$ gives $$g(1,0) = m$$
Added: note that they are not saying anything about $$\gcd(m,z,t).$$ The forms that are in (and contribute to $$h(d)$$) the form class group are primitive, but this problem may produce an imprimitive form. For that matter, they do not prohibit $$d$$ from being a square, which allows sort of degenerate forms that factor, such as $$x^2 - y^2$$