Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 11G / Solution

A form $a{x}^2+bxy+c{y}^2$ is positive definite when $a>0$ and $d=b^2-4ac<0$; equivalence is change by $S{L}_2(\mathbb{Z})$, and reduced means $|b|\le a\le c$ with the standard boundary sign convention. Translating $x\mapsto x+my$ reduces $|b|$, and swapping variables reduces $a$; discriminant bounds force termination. For a reduced form, $|b|\le a\le \sqrt{|d|/3}$ and $b\equiv d(\mathrm{mod}2)$. Discriminant $-28$ has one primitive reduced class, represented by $x^2+7y^2$. Therefore an odd prime is represented exactly when it splits in $\mathbb{Q}(\sqrt{-7})$, namely $(p/7)=1$, with the ramified exception $p=7$. Thus $p=7$ or $p\equiv 1,9,11,15,23,25(\mathrm{mod}28)$.

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