COORDINATES: Lectures meet Tues/Thur. 6:00--7:50 (#10290 01) in Science North 142.
Burton, David M. Elementary Number Theory (4th Ed.)
Andrews, George E. Number Theory Dover, 1994
WEB RESOURCE: 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.
GRADING: Your grade will be based on two exams, weekly quizzes, homework, participation and a portfolio of your work.
The breakdown of points is:
|Midterm||Final||Quizzes||Homework||Participation|| Portfolio |
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.
Please read the sections from the book listed before the date of the first lecture on that material. To encourage this, I will give credit for emailing me with questions and statements about the assigned reading (see "Participation" below).
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 small subset. Please write your solutions carefully.
You do not need to hand in the answers to question I've included in [square brackets], but you should do them to prepare for...
QUIZZES: There will be a short quiz at the end of class each Tuesday on the previous week's material. I will try to be very specific about what you should know, often by handing sheets of practice problems. Generally they will be comparable to easier problems I assign and to the examples given in the reading.
MIDTERM: Thursday, 8 February 2001 in class, Rearrange your schedule NOW if necessary.
FINAL: Tuesday, 13 March 2001 in class, Rearrange your schedule NOW if necessary.
PARTICIPATION: I expect you to generally show up prepared for class and willing to work. Please read the section(s) to be covered by the day before class, and send email to firstname.lastname@example.org with at least five (5) statements or questions about the reading. 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 five sentences that describe the main points of the reading. Twelve such emails over the course of the quarter will count as full credit. (Note that you need not send email before Review or Test Days.)
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 books (which I don't recommend).
MATERIAL: 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 hilights 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.
Our goal is to cover most of chapters 1--9 of Burton, culminating in Gauss's law of quadratic reciprocity. Other topics as time permits. I will continue to update the schedule below as the term progresses.
|3600 LECTURE AND ASSIGNMENT SCHEDULE|
|§ 2.1: Division Algorithm||1/2 T||p. 19: 2,3,5,8||1/11|
|§ 2.2 The GCD||1/2 T||p.25: 2ad,3||1/11|
|§ 2.3: Euclid's Algorithm||1/2 T||p.31: 1,2abc,4ac,7--9||1/11|
|§ 2.4: ax + by = c||1/4||p. 37:1,2,3,7||1/11|
|§ Week 2 Problems||1/9 T||#1-10||1/18|
|§ 1.1: Induction||1/9--11||Read on your own||1/16|
|§ Week 2 Problems||1/11 R||#1-10||1/18|
|§ Week 2 Probs (cont.)||1/16||1/18|
|Euclid's Lemma,Primes||1/18||p.43: 4,5,7,11,12; p.49: 1-3||1/25|
|§ 3.1-3: Primes||1/23||p.49: 12b,13a, 14; p. 58: 2,3||2/1|
|§ 4.1-2: Mods||1/25||p.67: 2,4,8,9;||2/1|
|§ 4.3: Divisibility tests||1/30||p.72: 6b,8,10,11||2/8|
|§ 4.4: CRT||1/30||p. 79:1--5,8||2/8|
|REVIEW FOR MIDTERM||2/6 T||(Attempt Practice Midterm by today)||.|
|MIDTERM EXAM: 2/8 R|
|§ 5.1-3: Fermat's Thm.||2/13||p. 93:1,4ab,12||2/23|
|§ 5.4: Wilson's Thm.||2/13||2/23|
|§ 6.1: Tau & Sigma||2/13||p.107: 2,3,5b,7,8,9;||2/23|
|§ 6.3: Floor fn.||2/15||p.119: #2bc,3,5,6||2/23|
|§ 7.1-2: Phi (Totient fn.)||2/15||p.127: 1,2,4ab||2/23|
|§ 7.3: Euler's Thm||2/20||p.127: 8,9a,11a,13; p.132: 1a,4,6,7,9;||3/1|
|§ 8.1: Orders||2/20||p.153: 1a,3,10,12||3/1|
|§ 8.2: Generators||2/20||p.160: 2,3,4a,10;||3/1|
|§ 8.4: Indices||2/22||p.169:2bc,3ad,12,17||3/1|
|§ 9.1: Euler's Criterion||2/27||p.175: 1b,3,4,7||3/8|
|§ 9.2: Legendre Symbol||2/27||p.186: #1abc,2abc,5;||3/8|
|§ 9.3: Quadratic Reciprocity||3/1||p.193: #1abc,3,5,14||3/8|
|§ 9.4: Composite mods||3/1||p.198: #2ab,4||3/8|
|REVIEW DAY||3/8 R|
|FINAL EXAM 3/13 T|
What are the next two terms of the following sequence: 0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, ---, --- ?
No Quiz on Tuesday 2/6 (except Take-home quiz due) Quiz #6 on Tuesday 6/13.
For the question about which U(m) are cyclic, you may find the following link to one of Joe Gallian's applets useful. It will let you quickly test any values of m you want.
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