3320prelim - 2 dy 2 = 1 is (x,y) = (p,q) , where d is an...

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Math 3320 Prelim Fall 2009 No notes, books, calculators or other electronic devices are allowed during this exam. 1. Determine how a 37 × 8 rectangle can be decomposed into 9 non-overlapping squares (of various sizes), and draw a Fgure showing how the 9 squares are arranged in the rectangle. 2. ±ind two di²erent linear fractional transformations that take the edge a 1 / 2 , 1 / 1 A in the ±arey diagram to the edge a 5 / 3 , 3 / 2 A (with 1 / 2 going to 5 / 3 and 1 / 1 going to 3 / 2). 3. Determine the value of the continued fraction 1 u ր 1 + 1 u ր 1 + 1 u ր 1 + 1 u ր 4 . 4. (a) ±ind the continued fraction for r 3 / 5. (b) ±ind the continued fractions for ( 5 + 17 )/ 2 and ( 5 17 )/ 2. 5. (a) ±or which integer values of n in the range 1 n 50 does the equation x 2 33 y 2 = n have integer solutions? (b) ±ind the smallest positive integer solution of x 2 33 y 2 = 1. (c) Suppose the smallest positive integer solution of the equation x
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Unformatted text preview: 2 dy 2 = 1 is (x,y) = (p,q) , where d is an arbitrary Fxed positive integer that is not a square. ind a formula for the second smallest solution, a formula in terms of d , p , and q . 6. (a) Suppose the topograph of a quadratic form Q(x,y) has a vertex where the values of Q in the three regions surrounding this vertex are numbers p , q , and r . ind a formula for the discrimi-nant of Q in terms of p , q , and r . p q r (b) Suppose the values of a quadratic form Q at the two ends of an edge in the topograph of Q are numbers p and q . Show that p and q have the same parity (i.e., both are odd or both are even). 7. Using a topograph, Fnd the quadratic form Q(x,y) satisfying the conditions that Q( 3 , 5 ) = 1, Q( 4 , 7 ) = 1, and Q( 7 , 12 ) = 1....
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