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Let x be an irrational number with convergents Pnf qn. For every n :'.::: 0, verify the

Elementary Number Theory | 7th Edition | ISBN: 9780073383149 | Authors: Professor David Burton ISBN: 9780073383149 413

Solution for problem 13 Chapter 15

Elementary Number Theory | 7th Edition

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Elementary Number Theory | 7th Edition | ISBN: 9780073383149 | Authors: Professor David Burton

Elementary Number Theory | 7th Edition

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Problem 13

Let x be an irrational number with convergents Pnf qn. For every n :'.::: 0, verify the following:1 I Pn I 1(a) < x - - < .2qnqn+l qn qnqn+l(b) The convergents are successively closer to x in the sense that[Hint: Rewrite the relationI Pn I I Pn-1 x-- < x---Iqn qn-1Xn+lPn + Pn-1x= ------Xn+lqn + qn-1as Xn+1(xqn - Pn) = -qn-1(x - Pn-ifqn-1).]

Step-by-Step Solution:
Step 1 of 3

- -1- _ Jgk _ A hjje(.,oC_m'-tc

Step 2 of 3

Chapter 15, Problem 13 is Solved
Step 3 of 3

Textbook: Elementary Number Theory
Edition: 7
Author: Professor David Burton
ISBN: 9780073383149

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Let x be an irrational number with convergents Pnf qn. For every n :'.::: 0, verify the