Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 1G / c / Solution

Every proper equivalence class of a positive definite form contains a unique reduced form $[a,b,c]$, wherewith $b\ge 0$ when equality occurs in either inequality. Its discriminant isFor a reduced positive definite form, $3a^2\le 80$, so $1\le a\le 5$. Also $b$ is even andChecking the finitely many possibilities with $|b|\le a\le c$ givesEach coefficient triple has greatest common divisor one. By uniqueness of reduced representatives, these are four distinct classes. Hence the requested number is