Welcome to Tom Roby's Math 3600 homepage! (Winter 2004)

(last updated: 9 March 2004)

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.

You may well initially find the unusual style of this course challenging and even frustrating. By persevering, you will gain mathematical maturity and develop resources that will be useful to you in the future, whatever path your career takes. Don't be shy about seeking help from me, your peers, or other resources if you need it!

CONTENT GOALS:

• Learn to do arithmetic and high school algebra in various mods including to
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 (mod m)
6. know under what conditions the above equations will have solutions and whether they will be unique or multiple.
7. Take powers and logarithms (mod 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 Theorem of Arithmetic
5. number theoretic functions: tau, sigma, phi

COORDINATES: Lectures meet Tues/Thur. 4:00--5:50 (#11978 01) in Science North 321.

TEXTS:

1. Silverman, Joseph H. A Friendly Introduction to Number Theory (2th Ed.) Prentice-Hall, 2001.
2. LeVeque, William J. Elementary Theory of Numbers Dover Publications, 1990 (orig. 1962).
3. LeVeque, William J. Fundamentals of Number Theory Dover Publications, 1996 (orig. 1977).

WEB RESOURCES: This course has a blackboard homepage. with three discussion forums 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://bb3.csuhayward.edu/ For help with blackboard email the ICS Help Desk at helpdesk@csuhayward.edu or call them at (510) 885-HELP. The homepage for this course is http://seki.csuhayward.edu/3600.html. 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 policy. It also is a poor learning strategy.

3600 LECTURE AND ASSIGNMENT SCHEDULE
Class Date Topics Read by Today Homework Due
1/6 T Overview, Divisibility in Z A1-9 1/15 R
1/8 R Arithmetic in Mods Lab Silv. 1,2,4 B1-9 1/15 R
1/13 T Euclid's Algorithm & Z[i] Silv. 5,6 A10, B10-12, C1-6 1/22 R
1/15 R Solving equations in mods D1,4-8 1/22 R
1/20 T ax+by=c; Magic box; U(n) Silv. 8, 33 E1-10 1/29 R
1/22 R Powers & Logarithms Silv. 7, 12 F1-7 1/29 R
1/27 T Primes, FTA Silv. 13,14 G1-8 2/10 T
1/29 R More on Primes Silv. 9-10 H1-11 2/10 T
2/3 T Thms of Fermat & Wilson; CRT Silv. 11,15 2/12 R
2/5 R REVIEW DAY (Attempt Practice Midterm by today) .
2/10 TUESDAY: MIDTERM EXAM
2/12 Tau, Sigma, Phi, perfect numbers I 1-9 2/19 R
2/17 Multiplicative functions Silv. 16, 19; Burt. 6.1 J 2/26 R
2/19 Cryptograpy: RSA Silv. 17,18 K 2/26 R
2/24 Orders & Generators Silv. 20-21 L1-6 3/4 R
2/26 Squares modulo p Silv. 22-23 L7-12 3/4 R
3/2 Quadratic Residues Silv. 24 M1-6 3/11 R
3/4 Quadratic Reciprocity Burton 9.3; Silv. 36 N1-7 3/11 R
3/9 Binomial Coeff.. CATCHUP DAY [Check website] O a-c 3/18 R
3/11 R REVIEW DAY (Attempt Practice Final by today)
3/18 THURSDAY: FINAL EXAM

• The Prime Pages includes numbers, research, and records. The Curois page includes a prime that is illegal and other numerology.

NEWS & NOTES

Here is the Practice Midterm.

Here is the Practice Final.

Here are the Practice Final Solutions.

What are the next two terms of the following sequence: 0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, ---, --- ?

[From Macalster College's Problem of the Week: A 0-1 Decision] How many primes are there that, in the usual base 10 notation, begin and end with a "1" and have alternating "0"s and "1"s?