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 discriminant 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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 '08
 LOZANOROBLEDO
 Math, Algebra, Number Theory, Greatest common divisor, Euclidean algorithm, Continued fraction, smallest positive integer

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