# Introduction

Number theory is the study of integers, primarily their structure under the operations of multiplication and addition. Though this seems a humble beginning, it is surprising how quickly one can ask mathematical questions about integers which are exceedingly difficult to resolve. Indeed, there are numerous problems which are thousands of years old that have yet to be resolved, as well as many other ancient problems which have only been resolved due to very sophisticated mathematics. It is this contrast between simplicity and complexity which forms the aesthetic of many number theoretic problems. In this class we'll develop some basic tools which allow us to begin analyzing the structures which govern the integers.

# Basic Information on the Course

## Webpage

Nearly all of the materials for this class are available at the course Wiki, http://math453spring2009.wikidot.com. Your participation in the development of the Wiki will be an important part of your grade, as outlined in the Idea Journal and Group Projects sections.

## Instructor

The professor for this course — Andy Schultz — is available to talk to you about anything you'd like, pretty much whenever you'd like. His office is Room 238 of Illini Hall, and his office telephone is 217-333-9860. He checks his email (moc.liamg|ztluhcs.c.werdna#moc.liamg|ztluhcs.c.werdna) somewhat obsessively, so this is probably the best way to get in touch with him.

Formal office hours are scheduled on

- Mondays from 10 to 11 and
- Tuesdays from 11 to noon
- Wednesdays from 3:30 to 4:15
- Thursdays from 2 to 3

but if these times don't work for you, feel free to set up an appointment via email.

## Text

The text for the course is Strayer's *Elementary Number Theory,* which has ISBN number 1-57766-224-5. You are expected to have a copy of this text once the course begins; if for some reason the bookstore is out of copies, moc.liamg|ztluhcs.c.werdna#ydnA liame and let him know.

# Grading

Your final grade will be determined based on the following breakdown

- 15% Homework
- 20% Midterm 1
- 20% Midterm 2
- 30% Final Exam
- 5% Participation
- 10% Group Project

If it becomes necessary, I also reserve the right to institute and enforce attendance policies for students whose average is C or below.

## Homework

Mathematics is best learned by doing, so part of your responsibility in this class is to complete weekly homework assignments. The basic rules for submitting homework are:

- Your assignment should reflect your own understanding.
- Your solutions should be written so that anyone with a basic knowledge of the course can read and understand them.
- Your solutions should include complete justification for your claims; unsubstantiated answers will receive no credit.
- Your assignment should be turned in at the beginning of class on the day it is due.

Collaboration is also a real part of how mathematics gets done "in the real world," and so you shouldn't interpret the first rule as meaning that you can't discuss problems with anyone. Indeed, you are encouraged to work on problems with your classmates or to talk to the instructor for help when you are stuck. If you do work in collaboration with someone else, though, there are two important rules to observe

- You should indicate on your paper with whom you have collaborated
- If you take notes while collaborating, you must
**destroy**these notes before starting to write up your own solution.

Failure to observe these rules when you've collaborated will result in all members of the collaboration getting a 0 on the submitted assignment.

## Tests

There will be two midterms in the course and a final. The midterms are scheduled for February 25th and April 10th; if these dates change for some reason, you will receive at least 2 weeks notice. The final is scheduled for Tuesday, May 12^{th}, from 7 to 10pm Wednesday, May 13^{th} from 7:00–10:00 PM. The University has very strict policies on rescheduling a final, so if you need to reschedule your midterm, be sure to contact Andy immediately.

## Idea Journal

As a means for creatively engaging with the course material, you'll be expected to make regular posts to the forum section of the course Wiki to record number theoretic ideas or conjectures which you formulate. Acceptable posts include anything which is relevant to the course material being discussed. This could mean that you're taking a known theorem and asking for a generalization in a particular direction or that you're posting a link to some other webpage which covers some interesting idea connected to the work we've done.

For instance, after reading Goldbach's conjecture

Goldbach's conjecture: Every even integer greater than 2 can be expressed as the sum of two (not necessarily distinct) prime numbers.

You might wonder: who is this guy Goldbach? This might lead you to Googling the name and reporting back a few interesting details. Alternatively, you might read Goldbach's conjecture and wonder: why is this limited to even numbers? This might lead you to trying to come up with a reasonable restatement of Goldbach's conjecture that includes odd numbers, or perhaps you'll give an explanation for why such a restatement will never be reasonable. Remember: the point is to creatively engage with the course material. Also remember that whenever you post, you should write clearly and give other class members a reasonable context to understand your post.

Your grade for the idea journal will be based on the frequency, content and quality of your posts. It should not be challenging for each student create one post in each chapter, nor should it be overly burdensome for each student to respond to one or two other posts per chapter. This basic level of participation will give you 75 of the possible 100 points on the journal.

## Group Projects

By the end of the term you will have enough number theoretic knowledge that you can explore some ideas on your own. To finish the semester, we'll split into small groups that are each responsible for understanding a particular corner of number theory. Each group will then be asked to give an in-class presentation on their work, as well as to post a page to the Wiki giving a good description of what they discovered in their project. Groups and topics will be determined later in the semester, but students are encouraged to think of topics they'd like to explore as soon as possible.