Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 1G (Number Theory)

The linear Diophantine equation solvability criterion says that $ax+by=c$ has an integer solution exactly when $\gcd (a,b)$ divides $c$. The Euclidean algorithm givesThus $\gcd (1007,2314)=1$, which divides $37$. Therefore

Applying the Euclidean algorithm,Hence $\gcd (2508,3211)=19$. Since $19$ does not divide $55$, the linear Diophantine equation solvability criterion gives

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.