Quadratic Mathematics | Part IB, 2001

Let q(x,y)=ax2+bxy+cy2q(x, y)=a x^{2}+b x y+c y^{2} be a binary quadratic form with integer coefficients. Define what is meant by the discriminant dd of qq, and show that qq is positive-definite if and only if a>0>da>0>d. Define what it means for the form qq to be reduced. For any integer d<0d<0, we define the class number h(d)h(d) to be the number of positive-definite reduced binary quadratic forms (with integer coefficients) with discriminant dd. Show that h(d)h(d) is always finite (for negative d)d). Find h(39)h(-39), and exhibit the corresponding reduced forms.

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