State what it means for two binary quadratic forms to be equivalent, and define the class number $h(d)$.

Let $m$ be a positive integer, and let $f$ be a binary quadratic form. Show that $f$ properly represents $m$ if and only if $f$ is equivalent to a binary quadratic form

$$m x^{2}+b x y+c y^{2}$$

for some integers $b$ and $c$.

Let $d<0$ be an integer such that $d \equiv 0$ or $1 \bmod 4$. Show that $m$ is properly represented by some binary quadratic form of discriminant $d$ if and only if $d$ is a square modulo $4 m$.

Fix a positive integer $A \geqslant 2$. Show that $n^{2}+n+A$ is composite for some integer $n$ such that $0 \leqslant n \leqslant A-2$ if and only if $d=1-4 A$ is a square modulo $4 p$ for some prime $p<A$.

Deduce that $h(1-4 A)=1$ if and only if $n^{2}+n+A$ is prime for all $n=0,1, \ldots, A-2$.