Unformatted text preview: Math 3320 Problem Set 5 1 For each of the ﬁrst six problems, compute the value of the given periodic or eventually periodic continued fraction by ﬁrst drawing the associated inﬁnite strip of triangles, then ﬁnding a linear fractional transformation T in LF (Z) that gives the periodicity in the strip, then solving T (z) = z . 1. 2. 3.
1 1 1 ր + 1ր 2 5 ր + 1ր + 1ր 2 1 1 ր + 1ր + 1ր + 1ր + 1ր + 1ր 1 1 1 1 1 2 4. 2 + 1ր + 1ր + 1ր 1 1 4 5. 2 + 1ր + 1ր + 1ր + 1ր 1 1 1 4 6.
1 ր + 1ր + 1ր + 1ր 1 1 2 3 7. Draw the topograph for the form Q(x, y) = 2x 2 + 5y 2 , showing all the values of Q(x, y) ≤ 60 in the topograph, with the associated fractional labels x/y . If there is symmetry in the topograph, you only need to draw one half of the topograph and state that the other half is symmetric. 8. Do the same for the form Q(x, y) = 2x 2 + xy + 2y 2 , in this case displaying all values Q(x, y) ≤ 40 in the topograph. 9. Do the same for the form Q(x, y) = x 2 − y 2 , showing all the values between +30 and −30 in the topograph, but omitting the labels x/y this time. 10. For the form Q(x, y) = 2x 2 − xy + 3y 2 do the following: (a) Draw the topograph, showing all the values Q(x, y) ≤ 30 in the topograph, and including the labels x/y . (b) List all the values Q(x, y) ≤ 30 in order, including the values when the pair (x, y) is not primitive. (c) Find all the integer solutions of Q(x, y) = 24 , both primitive and nonprimitive. (And don’t forget that quadratic forms always satisfy Q(x, y) = Q(−x, −y) .) ...
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