MTH 382 Spring 2003
Number Theory

## Instructor:  Dr. Nancy Eaton

E-mail me:  eaton@math.uri.edu
Phone:  874-4439
Office: Rm 222, Tyler Hall
Office hours: Monday 2:00-5:00, or Tuesday by apt.
Visit my web page:  WEB PAGE

Section 01-Kingston
Meets MWF: 9:00-9:50, Craw 221

Homework
Solutions to selected homework problems
Practice problems and Keys to exams
Maple
Portfolio

Students who require accommodations and who have documentation from
Disability Services (874-2098) should make arrangements with me as soon as possible.

Text:  Elementary Number Theory, by Kenneth H. Rosen, 4th edition.  There is a web site provided by the author to supplement the text.  www.awlonline.com/rosen

Course Content and Goals:
Number theory is the study of properties of the Integers, { ... , -3, -2, -1, 0, 1, 2, 3, ... } and the Natural Numbers, {1, 2, 3, 4, ...}. It is both elegant and beautiful.  It also has many interesting applications in computer science and cryptography.  These results can be understood and appreciated without much mathematical sophistication.  In this course, you learn to prove theorems as we explore ideas and topics.  We will cover the basics: sequences, divisiblity, primes, factoring, greatest common divisors, congruences, Linear Diophantine Equations and Möbius Inversion.  Other topics will be covered as time permits.

Motivation:  Some examples from the exercises.

Example:
Show that if a is an integer, then 3 divides a3- a.

Example:
The lucky numbers are generated by the following sieving process:  Start with the positive integers.  Begin the process by crossing out every second integer in the list, starting your count with the integer 1.  Other thatn 1, the smallest integer not crossed out is 3, so we continue by crossing out every third integer left, starting the count with the integer 1.  The next integer left is 7, so we cross out every seventh integer left.  Continue this process, where at each stage we cross out every kth integer left, where k is the smallest integer not crossed out, other than 1, not yet used in the sieving process.  The integers that remain are the lucky numbers.

Find all lucky numbers less than 100.

Show that there are infinitely many lucky numbers.

Example:
A Chinese puzzle found in the sixth century work of mathematician Chang Ch'iu-chien, called the "hundred fowls" problem, asks:  If a cock is worth five coins, a hen three coins, and three chickens together are worth one coin, how many cocks, hens, and chickens, totaling 100, can be bought for 100 coins?

I assign many problems from the book.  You are expected to do them all.  We will go over some of the solutions in class.  I will select from those assigned a few to hand in.  I will grade them and return them with written comments on your work.  See working together.  I will give you opportunities in class to put your work on the board.  Some homework problems will be given from the technology section.  For that you will need to use Maple or some other computer algebra system.  I will also give some assignments involving the web site:  www.awlonline.com/rosen.

From those exercises collected and returned to you, I will select 10 for you to rewrite and hand in as your portfolio for the class.  This is intended to be a collection of well thought out proofs from the course.  See working together.

Exams: (Tests 25%, Final 25%)
There will be two one-hour tests on Wednesday, March 5th and Wednesday, April 23rd and a final exam on Friday, May 9, 8:00 - 11:00 AM.  The exams will be taken from material that we covered in class, so take careful notes.

Sections Covered:
We will draw material from the following chapters in the book.

CHAPTER& TOPIC
1   The Integers
3   Primes and Greatest Common Divisors
4   Congruences
6   Some special congruences
7   Multiplicative Functions
Others as time permits.

Dates to remember:
Exam 1:    Wed. March 5
Exam 2:    Wed. April 23
Portfolio due:  Fri. May 2
Final Exam:  Fri. May 9, 8:00-11:00 AM

Maple:   To complete the maple assignments, you must use a computer that Maple installed on it.  All computers in computer labs on campus have Maple installed on them.  The Math Dept. computer lab in Tyler Hall, Rm 101 also has assistants to help you.  Click here to get the hours that it is open:  Tyler 101.  If you have never used Maple before, you might want to do the introductory worksheet:  Download "Introduction to Maple"(intro141.mws)

Homework assignments: Check this list frequently to get the latest homework assignments.  It will be updated as we go.  We will go over some of these in class, and I will ask that you hand some in.

To get started, for friday Jan 24, DO:   1.1, #8;   1.3, #10,29;  3.1, #22;  4.1, #15  These are also listed below.
 Chapter Section Exercises Hand In Due # Chapt 1 1.1 3, 7, 8, 9, 13, 14, 15, 16 14 2/5 1 1.2 1, 2, 6, 7, 8, 13, 14, 20, 26, 27 8 2/5 1 1.3 1, 3, 10, 29 Handout: 1,2 2/5 1 1.4 1-13, 16, 19, 21-23, 33, 34,  39, 40, 48 22 2/5 1 Appendix A Page 517 1-8 2a, 6a 2/18 2 Chapt 3 3.1 1, 3, 6, 9, 10, 12, 13, 14, 17, 22, 24 Computations: 5, 6 10, 14, 24 Computations6 (maple) 2/26 3 3.2 1, 3, 4, 5, 6, 7, 8, 15, 16, 22, 23 8, 16, 22 3/3 4 Accepting late work and retries for Last chance See * below 3/17 A 3.3 1, 2, 3, 4 Computations: 2, using numbers in 1 2, 4  Computations2 (maple) 3/21 5 3.4 1, 5-13, 31-33, 35-40, 43, 48-50, 62-66, 69-71, Computations: 2 8, 10, 38, 40 Computations2 (maple) 3/31 6 3.5 1-4, 17 17 4/15 7 3.6 1-6, 10-13, 21 2 4/15 7 Chapt 4 4.1 1, 2, 4-15, 18, 22, 23, 26, 28 4, 8, 22 4/15 7 extra credit extra credit do only 5 4/9 EC 4.2 1-3, 5-9 4.3 1,2,3,4,7,8,9 2,4,8 Chapt 6 6.1 1-7 all 6.2 6.3 Chapt 7 7.1 7.2 7.3

* A for Amnesty.  This is your last chance to hand in any of the problems that were due for homeworks 1, 2, 3, or 4.   Also if you turned it in already and did not get full credit, you can redo the problem and hand it in again.  PLEASE - clearly mark the Homework number, section number and problem number.  Also, you must turn in the original so that I can see how many points I gave you the first time.

Solutions to selected homework problems and examples.

Practice problems and keys to Exams:

Portfolio:

I will have office hours that day from 2 to 5pm on May 2.   Either hand it in that day, or stop by my office to ask questions on the ones that you are not sure about and then hand it in on May 5.

Give complete questions to each problem.  Write the solution as formally as you can.  Use complete sentences.  Write it as if it would be published.  In fact, I will publish selected solutions on the web!  Either type or write very neatly.  If you type, it is okay to type the sentences and leave spaces for writing equations by hand.  If you have access to an equation editor, that would be best.

Selected problems for the portfolio:  1.2: 8;  1.4: 22;  Appendix 6a, 3.1: 10, 24;  3.2: 8;  3.4: 10, 38;  4.1:  22; Prove that the square root of 2 is irrational.

Put the problems in the above order and if you work with others, follow the guide below.

Working together:
I encourage you to work together under the following
circumstances.

1. Begin the problem on your own, do as much as you can.