2LECTURE 1.BINARY QUADRATIC FORMSLetx=m1v+m2w∈Λ. The length ofxis given by the formulax||2=||m1v+m2w||2= (m1, m2)v·vv·wv·ww·wm1m2=am21+ 2bm1m2+cm22,wherea=v·v,b=v·w,c=w·w.(1.1)Let us consider the (binary) quadratic form (thedistance quadratic formof Λ)f=ax2+ 2bxy+cy2.Notice that its discriminant satisfiesD= 4(b2-ac) =-4A(v,w)2<0.(1.2)Thusfis positive definite. Given a positive integernone may ask about integralsolutions of the equationf(x, y) =n.If there is an integral solution (m1, m2) of this equation, we say that the binaryformfrepresentsthe numbern.Geometrically this means that the circle ofradius√ncentered at the origin contains one of the pointsx=m1v+m2wof the lattice Λ. Notice that the solution of this problem depends only on the
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Quadratic form, Binary Quadratic Forms, binary quadratic form, distance quadratic form