Mathematics — Number Theory · zero-hyperbolic-parabolic-forms
Problem
(a) Show that if a quadratic form Q(x,y)=ax2+bxy+cy2 can be factored as a product (Ax+By)(Cx+Dy) with A,B,C,D integers, then Q takes the value 0 at some pair of integers (x,y)=(0,0), hence Q must be either 0-hyperbolic or parabolic. Show also, by a direct calculation, that the discriminant of this form is a square.
(b) Find a 0-hyperbolic form Q(x,y) such that Q(1,5)=0 and Q(7,2)=0 and draw a portion of the topograph of Q that includes the two regions where Q(x,y)=0.