Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 1I / Solution

A reduced positive definite binary quadratic form $f=[a,b,c]=a{x}^2+bxy+c{y}^2$ satisfieswith $b\ge 0$ when $|b|=a$ or $a=c$. Its discriminant of a binary quadratic form is $d=b^2-4ac<0$. The first displayed inequality already gives $|b|\le a$, while $c\ge a$ givesTaking the positive square root yieldsMoreover, $d=b^2-4ac\equiv b^2\equiv b(\mathrm{mod}2)$, so $b\equiv d(\mathrm{mod}2)$. For fixed negative $d$, the displayed bounds leave only finitely many possible integers $a$ and $b$, and then $c=(b^2-d)/(4a)$ is uniquely determined. Hence only finitely many reduced forms have discriminant $d$.

For $d=-15$, the bound gives $a\le \sqrt{5}$, so $a\in \{1,2\}$. The modular congruence condition makes $b$ odd. If $a=1$, then $b=\pm 1$ and $c=4$; the boundary convention $|b|=a\Rightarrow b\ge 0$ retains $[1,1,4]$. If $a=2$, then $b=\pm 1$ and $c=2$; the boundary convention $a=c\Rightarrow b\ge 0$ retains $[2,1,2]$. Thus the complete list is