3320hw6 - the separator line and use this to nd a second...

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Math 3320 Problem Set 6 1 1. Determine the periodic separator line in the topograph for each of the following quadratic forms (you do not need to include the fractional labels x/y ): (a) x 2 - 7 y 2 (b) 3 x 2 - 4 y 2 (c) x 2 + xy - y 2 2. Using your answers in the preceding problem, write down the continued fraction expansions for 7, 2 3 / 3, and ( - 1 + 5 )/ 2. 3. For the following quadratic forms, draw enough of the topograph, starting with the edge separating the 1 / 0 and 0 / 1 regions, to locate the periodic separator line, and include the separator line itself in your topograph. (a) x 2 + 3 xy + y 2 (b) 6 x 2 + 18 xy + 13 y 2 (c) 37 x 2 - 104 xy + 73 y 2 4. For the quadratic form x 2 - 14 y 2 do the following things: (a) Draw the separator line in the topograph and compute the continued fraction for 14. (b) Find the smallest positive integer solutions of x 2 - 14 y 2 = 1 and x 2 - 14 y 2 = - 1, if these equations have integer solutions. (c) Find the linear fractional transformation that gives the periodicity translation along
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Unformatted text preview: the separator line and use this to nd a second positive solution of x 2-14 y 2 = 1. (d) Determine the integers n with | n | 12 such that the equation x 2-14 y 2 = n has an integer solution. (Dont forget the possibility that there could be solutions (x,y) that arent primitive.) 5. For the quadratic form x 2-29 y 2 do the following things: (a) Draw the separator line and compute the continued fraction for 29. (b) Find the smallest positive integer solution of x 2-29 y 2 = -1. (c) Find a glide-reection symmetry of the separator line and use this to nd the smallest positive integer solution of x 2-29 y 2 = 1. 6. Compute the periodic separator line for the form x 2-43 y 2 and use this to nd the continued fraction for 43....
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