Project Title: p-Adic Number Theory and Continued Fractions

Notes/Pictures: · October 6, 2008 · October 13, 2008 · October 20, 2008 #1 · October 20, 2008 #2 · October 20, 2008 #3 · October 27, 2008 #1 · October 27, 2008 #2 · October 27, 2008 #3 · October 27, 2008 #4 · November 12, 2008 #1 · November 12, 2008 #2 · November 12, 2008 #3 · November 17, 2008 #1 · November 17, 2008 #2 · January 26, 2009 · February 4, 2009 #1 · February 4, 2009 #2 · February 4, 2009 #3 · February 12, 2009 · February 18, 2009 #1 · February 18, 2009 #2 · February 18, 2009 #3

Resources:

Mathematica Files:

· p-Adic Continued Fractions Calculator

Introductory Material:

· p-adic Number, MathWorld and p-adic Number, Wikipedia
The usual blanket of information to give you a flavor of the theory. MathWorld article is short and matter of fact. Wikipedia is longer, has more examples, and shows both the analytical and algebraic views. Both are a great place to start and take a glance at, but a horrible place to stop and delve into.

· A first introduction to p-adic numbers, David Madore
A good 5-page introduction to p-adics that provides some good ways to think about them concretely. A few good examples of the operations are shown.

· An Introduction to p-adic Numbers and p-adic Analysis, A.J. Baker
Chapter 1 is some familiar material from our Number Theory class, though you may not recognize it since the notation is a little funny. Chapter 2 gives the analytical background of the p-adics and gets pretty heavy towards the end. Chapter 3 covers the notations of convergence which we’ll definitely need. Chapters 4 & 5 are on topology and advanced topics and shouldn’t be important for what we want to do.

· Visualizing the p-adic Integers, A. Cuoco
Another good introduction, but haven’t really read through it all. It might be helpful to start 3 or 4 pages in. Looks like we shouldn’t have to worry about the pentagon and triangle pictures in what we’re doing, but they might be helpful in “visualizing” the p-adics.

· p-adic Numbers and Adeles - An Introduction, Matthew R. Watkins
Another overview, this time starting with valuations, a motivating idea for the p-adics. Goes a little beyond what we’re interested since we aren’t going to worry about the adeles or the implications in physics.

Advanced Material:

· Continued Fractions in Local Fields, Jerzy Browkin

· Continued Fractions in Local Fields II, Jerzy Browkin