### Ma458 Homework 5

Ethan Bolker
Fall 2013

This assignment is due next Thursday (10/15) but start it over the weekend so we can discuss it some on Tuesday.

Read Chapters 12 and 14 in Silverman. (Note that I've skipped 13. There's some really interesting stuff there, but it's mostly expository. There's little in it you can actually learn to do in this course.

1. Silverman, Exercise 12.2. Part (a) is easy once you really understand the proof of Theorem 12.2. When you write up your solution, try to copy Silverman's writing style.

Part (b) is easy once you understand Silverman's argument showing that Theorem 12.2 is false for primes congruent to 1 modulo 4.

2. Read Chapter 18 (yes, skip to 18) - to understand the general idea, not the details. We'll go back to Chapters 16 and 17 for the background we need for those details.

3. Be prepared to present two different proofs that there is no rational number whose square is 2. (Note that this statement never uses the words "square root". You proofs shouldn't either).

4. Use the strategy we developed in class today to study the diophantine equation
x2 - 3 y 2 = +- 1.

If you have the time and energy, think about

x2 - d y 2 = +- 1
for larger values of d.