Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 1F / Solution

A reduced positive definite binary quadratic form $[a,b,c]$ satisfieswith $b\ge 0$ when either $|b|=a$ or $a=c$. For $d\equiv 0$ or $3(\mathrm{mod}4)$, the class number of a negative discriminant $h(-d)$ is the number of proper equivalence classes of primitive positive definite integral forms of discriminant $-d$, equivalently the number of their reduced representatives.

WritePartitioning the $k$ complete prime-power factors between $r$ and $s$ gives $2^k$ ordered factorizations $d=rs$ with $\gcd (r,s)=1$. After identifying $(r,s)$ with $(s,r)$, there are $2^{k-1}$ choices with $r\le s$. Each gives the primitive reduced formwhose discriminant is $-4rs=-4d$. The uniqueness of reduced representatives makes these classes distinct, proving the coprime-factorization lower bound for a quadratic-form class number

The inequality can be strict. For $d=5$, there is one distinct prime factor, while the reduced primitive forms of discriminant $-20$ areThus $h(-20)=2>1=2^{1-1}$.

Solved by gpt-5.6-sol high.