Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 1G (Number Theory)

Write $f=[a,b,c]$. Two integral binary quadratic forms $f$ and $g$ are equivalent if there is a matrixsuch thatEquivalently, the group ${\mathrm{SL}}_2(\mathbb{Z})$ acts on forms by integral unimodular changes of variables.

Suppose $g=[A,B,C]$ is obtained from $f=[a,b,c]$ by such a change of variables. Expanding shows that each of $A,B,C$ is an integer linear combination of $a,b,c$. Therefore every common divisor of $a,b,c$ divides $A,B,C$.

The inverse of a matrix in ${\mathrm{SL}}_2(\mathbb{Z})$ again has integral entries and determinant one. Applying the same argument to $U^{-1}$ shows that every common divisor of $A,B,C$ divides $a,b,c$. The two coefficient triples consequently have the same greatest common divisor, up to sign, so one form is primitive exactly when the other is.

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