Topology of Numbers · Problem 6.2.1

The key is understanding that a quadratic form $ax^2 + bxy + cy^2$ with discriminant $\Delta = b^2 - 4ac$ can represent an integer $n$ only if $\Delta$ is a quadratic residue modulo $4n$. For $n = 5$, you need $\Delta \equiv \square \pmod{20}$, which severely restricts which discriminants work. Once you identify valid $\Delta$ values, finding an explicit form boils down to solving $b^2 - 4ac = \Delta$ and checking whether integer solutions $(a, b, c)$ actually produce $5$ as a value—often by testing small inputs or using the theory of reduced forms in each class.

The key is understanding that a quadratic form $ax^2 + bxy + cy^2$ with discriminant $\Delta = b^2 - 4ac$ can represent an integer $n$ only if $\Delta$ is a quadratic residue modulo $4n$. For $n = 5$, you need $\Delta \equiv \square \pmod{20}$, which severely restricts which discriminants work. Once you identify valid $\Delta$ values, finding an explicit form boils down to solving $b^2 - 4ac = \Delta$ and checking whether integer solutions $(a, b, c)$ actually produce $5$ as a value—often by testing small inputs or using the theory of reduced forms in each class.