## Welcome to Tom Roby's Math 3600 homepage! (Winter 2003)

(last updated: 7 January 2003)

### Class Information

LEARNING GOALS: This course has an importance that goes beyond the content. Most students, even some math majors, think of mathematics as learning procedures and practicing them. It's easy for teachers and students to fall into this habit. Instead I want all students to develop the mathematical habits of thought that are necessary for doing and teaching mathematics. Specifically:

• Learn to experiment with examples, make conjectures, construct rigorous proofs and disproofs.
• Learn to persevere on hard problems and use resources (books, internet, colleagues) effectively.
• Experience mathematics as a dynamic human activity, rather than a set of unwavering recipes handed down on stone tablets.

CONTENT GOALS:

• Learn to do arithmetic and HS algebra in various mods including
1. add, subtract, multiply, divide, invert, take powers
2. solve linear equations in one variable (mod m)
3. solve systems of linear equations in one variable (mod m)
4. solve quadratic equations (mod m)
5. solve some higher power polynomial equations
6. know under what conditions the above equations will have solutions and whether they will be unique or multiple.
7. Take powers and logarithms (log p) and use them to solve equations.
• Gain a deeper understanding of the arithmetic of the integers including:
1. the division algorithm and Euclid's algorithm
2. solving linear diophantine equations
3. divisibility properties and divsibility tests
4. prime numbers and the Fundamental Thm. of Arithmetic
5. number theoretic functions: tau, sigma, phi

COORDINATES: Lectures meet Tues/Thur. 6:00--7:50 (#12043 01) in Science South 213.

TEXTS: (1) Silverman, Joseph H. A Friendly Introduction to Number Theory (2th Ed.) Prentice-Hall, 2001. (2) Davenport, H. The Higher Arithmetic (7th Ed.) Cambridge, 1999.

WEB RESOURCES: This course has a blackboard homepage. with a discussion forum that I strongly encourage you to use. Your participation grade will be partly based on the number of useful messages you contribute. Go to blackboard at http://bb1.csuhayward.edu/ For help with blackboard email the ICS Help Desk at helpdesk@csuhayward.edu or call them at (510) 885-HELP. http://seki.csuhayward.edu/3600.html will be my Math 3600 homepage. It will include a copy of the syllabus and list of homework assignments. I will keep this updated throughout the quarter.

The breakdown of points is:

Midterm Final Homework Participation Portfolio
25% 25% 20% 20% 10%

LEARNING: The only way to learn mathematics is by doing it! Complete each assignment to the best of your ability, and get help when you are confused. Come to class prepared with questions. Don't hesitate to seek help from other students. Sometimes the point of view of someone who has just figured something out can be the most helpful.

DISABILITIES If you have a documented disability and wish to discuss academic accommodations, or if you would need assistance in the event of an emergency, please contact me as soon as possible.

HOMEWORK: Homework will be given for each lecture, and all the homework assigned the previous week will be due the following Thursday. Please attempt all the problems by Tuesday, so that you can ask any questions you may have in class then. Except for routine computations, you should always give reasons to support your work and explain what you're doing. Not all the problems will be graded, but only a subset. Please write your solutions carefully.

The homework is meant to be challenging and somewhat open ended. You may find yourself spending lots of time working on them and sometimes getting frustrated. This is natural. I encourage you to work with other people in person and using blackboard. It's OK to get significant help from any resource, but in the end, please write your own solution in your own words. Copying someone else's work without credit is plagiarism and will be dealt with according to university policty It also is a poor learning strategy.

MIDTERM: Thursday, 11 February 2003 in class, Rearrange your schedule NOW if necessary.

FINAL: Tuesday, 18 March 2003 in class, Rearrange your schedule NOW if necessary.

PARTICIPATION: I expect you to generally show up prepared for class and willing to work. Please complete any assigned reading by by the day before class. Post at least 3 questions to blackboard or email them to 3600@seki.mcs.csuhayward.edu . This will help me focus classtime where you need it most. The questions can be anything from "What does the following sentence from the text mean..." to "Why is it important that the derivative measures the slope?" But please be as clear as possible about where the confusion is. Questions like "What's a prime number?" or "I don't understand the Euclidean Algorithm?" are less useful than "I don't understand why 1 is not considered a prime number" or "Here's my attempt to compute the gcd of 343 and 182 using Euclid's algorithm, but I seem to get the wrong answer...". If you don't have any questions, then come up with three sentences that describe the main points of the reading. Twelve such postings or emails over the course of the quarter will count as full credit. (Note that you need not send email before Review or Test Days.) Since this class is so large, maybe odd people should send on Tuesday and evens should send on Thursday (based on last digit of your student ID), but don't feel you need to stick to that rigorously.

PORTFOLIO: Please organize your work neatly in some sort of binder (e.g., 3-ring), so that you can refer to all your classnotes, homework assignments, quizzes, exams, handouts, and emails on the reading. I will check them at the end of the term. This will not only help you during the class, but also later when you want to recall something you learned but can't quite remember. It gives you a permanent record of what you learned even if you sell your book (which I don't recommend).

CONTENT: Number theory is one of the most beautiful subjects in all of mathematics. People's fascination with number paterns predates history and makes up the bulk of the knowledge that Euclid wrote down in his Elements . It has many problems that are easy to understand but fiendishly difficult to solve. Perhaps the most famous of these is "Fermat's Last Theorem", whose solution by Andrew Wiles after 350 years is one of the highlights of 20th century mathematics.

Many topics in number theory connect with and extend the pre-college mathematics curriculum in interesting ways. We will spend a lot of time solving equations, but the rules will be somewhat different than you are used to. This should help you gain a deeper understanding of why certain procedures you have learned in school (and may teach again later) work.

3600 LECTURE AND ASSIGNMENT SCHEDULE
Class Date Topics Homework Due
1/7 T Overview, Divisibility in Z A1-10 1/16 R
1/9 R Euclid's Algorithm & Z[i] B1-10 1/16 R
1/14 T Arithmetic in Mods Lab C1-11 1/23 R
1/16 R Solving equations in mods Lab D1,4-8 1/23 R
1/21 T ax+by=c; Magic box; U(n) C12, D2-3, E1-9 1/30 R
1/23 R Primes, FTA F1-7 1/30 R
1/28 T Powers & Logarithms G1-10 2/11 T
1/28 T More on Primes H1-10 2/11 T
2/4 R Theorems of Fermat & Wilson I1-10 2/13 R
2/6 R REVIEW DAY (Attempt Practice Midterm by today) .
2/11 T: MIDTERM EXAM
2/13--3/11 UNDER CONSTRUCTION
3/13 R REVIEW DAY (Attempt Practice Final by today)
3/18 T: FINAL EXAM