Paper 3, Section II, I

Number Theory
Part II, 2016

What does it mean for a positive definite binary quadratic form to be reduced?

Prove that every positive definite binary quadratic form is equivalent to a reduced form, and that there are only finitely many reduced forms with given discriminant.

State a criterion for a positive integer nn to be represented by a positive definite binary quadratic form with discriminant d<0d<0, and hence determine which primes pp are represented by x2+xy+7y2x^{2}+x y+7 y^{2}.