Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 1I (Number Theory)

Use the reduction algorithm for a positive definite binary quadratic form. A determinant-one shear $(x,y)\mapsto (x+ky,y)$ puts the middle coefficient in $[-a,a]$. If the resulting last coefficient is smaller than $a$, the determinant-one substitution $(x,y)\mapsto (-y,x)$ decreases the positive integer leading coefficient. Repeating terminates with a reduced positive definite binary quadratic form satisfying $|b|\le a\le c$. These substitutions preserve proper equivalence of binary quadratic forms and the discriminant of a binary quadratic form.

The enumeration of reduced binary quadratic forms givesSince $b^2\equiv -163\equiv 1(\mathrm{mod}4)$, $b$ is odd. For $|b|=1,3,5,7$, the possible values of $(b^2+163)/4$ are respectively $41,43,47,53$. Each is a prime number greater than seven. But $ac=(b^2+163)/4$, so its positive divisor $a\le 7$ must be one. Then $|b|\le a$ forces $b=\pm 1$ and $c=41$. The two signs are properly equivalent: substituting $x\mapsto x-y$ in $x^2+xy+41y^2$ gives $x^2-xy+41y^2$. Thus every form in question is properly equivalent to