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hw7 - its discriminant(b Find the source vertex or edge in...

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Math 3320 Problem Set 7 1 1. Find a hyperbolic quadratic form whose periodic separator line has the following pattern: 2. Use a quadratic form to compute continued fractions for the following pairs of numbers: (a) ( 3 + 6 )/ 2 and ( 3 6 )/ 2 (b) ( 11 + 13 )/ 6 and ( 11 13 )/ 6 (c) ( 14 + 7 )/ 9 and ( 14 7 )/ 9 3. (a) Find two 0-hyperbolic forms that have the same discriminant but take on dif- ferent sets of values. Draw enough of the topographs of the two forms to make it apparent that they do not have exactly the same sets of values. (Remember that the topograph only shows the values Q(x,y) for primitive pairs (x,y) .) (b) Do the same thing with two elliptic forms that take on positive values. Include the source vertex or source edge in the topographs. (c) Do the same thing with two hyperbolic forms, drawing their separator lines. 4. (a) Show the quadratic form Q(x,y) = 92 x 2 74 xy + 15 y 2 is elliptic by computing
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Unformatted text preview: its discriminant. (b) Find the source vertex or edge in the topograph of this form. (c) Using the topograph of this form, ±nd all the integer solutions of 92 x 2 − 74 xy + 15 y 2 = 60, and explain why your list of solutions is a complete list. (There are exactly four pairs of solutions ± (x,y) , three of which will be visible in the topograph.) 5. Show that if a quadratic form Q(x,y) = ax 2 + bxy + cy 2 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 (x,y) n= ( , ) , hence must be either 0-hyperbolic or parabolic. Show also, by a direct calculation, that the discriminant of this form is a square. (Do not quote the general fact that a quadratic form is 0-hyperbolic or parabolic if and only if it takes the value 0.)...
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