NT2009HW7

# NT2009HW7 - 4400/6400 PROBLEM SET 6 Recommendation 4400...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 10/26/2011 for the course MATH 4400 taught by Professor Staff during the Spring '11 term at UGA.

### Page1 / 2

NT2009HW7 - 4400/6400 PROBLEM SET 6 Recommendation 4400...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online