prob4

# prob4 - [Hint Add the relations p = 1 p 1 = 3 p k = k 1 p...

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MATH 115B PROBLEM SET IV MAY 10, 2007 (1) Express each of the rational numbers below as ﬁnite simple continued fractions: (a) - 19 / 51 (b) 187 / 57 (c) 71 / 55 (2) Determine the rational numbers represented by the following simple continued frac- tions: (a) [ - 2; 2 , 4 , 6 , 8] (b) [4; 2 , 1 , 3 , 1 , 2 , 4] (c) [0; 1 , 2 , 3 , 4 , 3 , 2 , 1] (3) If r = [ a 0 ; a 1 , a 2 , . . . , a n ], where r > 1, show that 1 r = [0; a 0 , a 1 , . . . , a n ] . (4) Represent the following simple continued fractions in an equivalent form, but with an odd number of partial denominators: (a) [0; 3 , 1 , 2 , 3] (b) [ - 1; 2 , 1 , 6 , 1] (4) Compute the convergents for the following simple continued fractions: (a) [1; 2 , 3 , 2 , 2 , 1] (b) [ - 3; 1 , 1 , 1 , 1 , 3] (5) By means of continued fractions, determine the general solutions of each of the fol- lowing diophantine equations: (a) 19 x + 51 y = 1. (b) 364 x + 227 y = 1. (6) (a) If C k = p k /q k is the k -th convergent of the simple continued fraction [1; 2 , 3 , 4 , ..., n, n + 1], show that p n = np n - 1 + np n - 2 + ( n - 1) p n - 3 + ··· + 3 p 1 + 2 p 0 + ( p 0 + 1)

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Unformatted text preview: . [Hint: Add the relations p = 1, p 1 = 3, p k = ( k + 1) p k-1 + p k-2 for k = 2 , ..., n .] (b) Illustrate part (a) by calculating the numerator p 4 for the fraction [1; 2 , 3 , 4 , 5]. (7) If C k = p k /q k is the k-th convergent of the simple continued fraction [ a ; a 1 , ..., a n ], show that q k ≥ 2 ( k-1) / 2 , 2 ≤ k ≤ n. [Hint: Observe that q k ≥ a k q k-1 + q k-2 ≥ 2 q k-2 .] 1 2 MATH 115B PROBLEM SET IV MAY 10, 2007 (8) If C k = p k /q k is the k-th convergent of the simple continued fraction [ a ; a 1 , ..., a n ], and a > 0, show that p k p k-1 = [ a k ; a k-1 , ..., a 1 , a ] , and q k q k-1 = [ a k ; a k-1 , ..., a 2 , a 1 ] , Hint: In the ﬁrst case, notice that p k p k-1 = a k + p k-2 p k-1 = a k + 1 p k-1 p k-2 ....
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prob4 - [Hint Add the relations p = 1 p 1 = 3 p k = k 1 p...

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