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Unformatted text preview: 3 and hence a b c d = a b c d . This can be also expressed by saying that the form f is obtained from the form f by using the change of variables x x + y, y x + y. We write this in the form f = Mf. According to Lagrange two binary quadratic forms f and g are called equivalent if one transforms to another under the change of variables as above defined by an integral matrix with determinant 1. An equivalence class is called the class of forms . Obviously, for any n Z , the set of integral solutions of the equations f ( x,y ) = n depends only on the class of forms to which f belongs. Also it is clear that two equivalent forms have the same discriminant. 1.2 As we saw before any lattice determines a class of forms expressing the distance from a point in to the origin. Conversely, given a positive definite binary form f = ax 2 + 2 bxy + cy 2 we can find a lattice corresponding to this form. To do this we choose any vector v of length...
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
- Fall '09