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Unformatted text preview: 4400/6400 PROBLEM SET 6 Recommendation: 4400 students should do at least three problems; 6400 students should do at least four. Heads up: There is an online applet for finding the fundamental solution to a Pell equation, available at http://www.numbertheory.org/php/pell.html . When asked for the fundamental solution u to a Pell equation you can use this applet, learn about continued fractions and try to find the solution yourself, or do some combination of the two. 6.0) Prove Lemma 1 of [Pell’s Equation]: let ( x,y ) be a nontrivial integral so- lution to x 2- Dy 2 = 1. Then: (i) x,y > ⇐⇒ x + √ Dy > 1. (ii) x > , y < ⇐⇒ < x + √ Dy < 1. (iii) x < , y > ⇐⇒ - 1 < x + √ Dy < 0. (iv) x,y < ⇐⇒ x + √ Dy <- 1. 6.1) Find all integral solutions to the following equations: a) x 2- 5 y 2 = 1. b) x 2- 53 y 2 = 1. c) x 2- 73 y 2 = 1. d) x 2- 1006009 y 2 = 1. 6.2) a) Show that for any nonsquare positive integer d and any positive integer M there exist infinitely many integral solutions to the Pell equation...
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This note was uploaded on 10/26/2011 for the course MATH 4400 taught by Professor Staff during the Spring '11 term at UGA.
- Spring '11
- Number Theory