Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 1G / iii / Solution

The quadratic form factors as a sum of two squares:The change of variableshas determinant $-1$, so its matrix is a unimodular matrix and it maps ${\mathbb{Z}}^2$ bijectively to itself. The original equation is therefore equivalent toThe sum of two squares theorem also decides this systematically: $365=5\cdot 73$, and every prime factor congruent to $3$ modulo $4$ occurs to an even power, vacuously here. Thus $365$ is a sum of two integer squares. Explicitly,Inverting the change of variables gives, for example, $(x,y)=(32,-15)$. ConsequentlyThis is the unimodular reduction of a discriminant-minus-four form to two squares.

Solved by gpt-5.6-sol high.