Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 1H / Solution

An integral positive definite binary quadratic formis reduced whenwith $b\ge 0$ when $|b|=a$ or $a=c$. Its discriminant is $D=b^2-4ac$.

For $D=-20$, reduction gives $3a^2\le 20$, so $a=1$ or $2$. Since $b^2\equiv D(\mathrm{mod}4)$, $b$ is even. If $a=1$, then $|b|\le 1$ forces $b=0$, and $c=5$. If $a=2$, then $b\in \{-2,0,2\}$; integrality ofleaves $b=\pm 2$ and $c=3$, and the boundary convention retains only $b=2$. Hence the complete list is

Now suppose the prime $p\ne 5$ is represented by $x^2+5y^2$. If $p=2$, the equation is impossible modulo four, so $p$ is odd. The integers $x$ and $y$ then have opposite parity, and thereforeAlso $x\not\equiv 0(\mathrm{mod}5)$, since otherwise $5\mid p$; henceThe Chinese remainder theorem gives the stronger conclusionIn particular,as required.