Northern Illinois University

DuSable Hall 302

Monday, Wednesday 4 - 5:15

Richard Blecksmith

**INTRODUCTION TO NUMBER THEORY (3 semester hours)**

Basic concepts in problem solving, methods of proof, divisibility, primes, integer sequences, number-theoretic functions, and selected topics.

Not used in major or minor GPA calculations for mathematical sciences majors or minors. It is a required course for the Minor in Mathematics Education.

**PREREQUISITE:**
MATH 229 or consent of department.

- Office: Watson Hall 355
- email: richard@math.niu.edu
- Phone: 753-6753
- Office Hours: Mon, Wed 3-4 and Wed 1-2

**SYLLABUS:**
The course will cover most of Modules 1-6 of the text.

- Module 1. Basic Concepts in Number Theory
- Definition of natural numbers and integers
- Divisibility
- Clock Arithmetic
- Congruences
- Congruence Facts
- Induction
- The Divisibility Algorithm
- Mod $n$ Tables
- The Pails of Water Problem
- Greatest Common Divisor
- Secret Codes

- Module 2. An Elementary Approach to Primes
- Primes and Composites
- Katie's Theorem
- Are the primes plentiful or scarce?
- Prime Deserts
- Euclid's Theorem
- The Prime Number Theorem
- Twin Primes
- Prime Constellations
- Cluster Primes

- Module 3. The Fundamental Theorem of Arithmetic
- Sieve of Eratosthenes
- Factoring into Primes
- Proof of Fundamental Theorem
- Applications of The Fundamental Theorem
- Primes of Special Type

- Module 4. Mediants, Contiguous Numbers, and Farey's Sequence
- Vicoria's Theorem
- Contiguous Fractions
- The Farey Sequence

- Module 5. Fibonacci Numbers
- Definition
- Worksheet

- Module 6. Partitions

**WITHDRAWAL:**
The last day for undergraduates to withdraw from a full-session course
is Friday, October 19.

**GRADING:**
Grades will be assigned on the basis of 400 points, as follows:

- 2 hour exams worth 100 points each
- Homework, attendance, writing assignments, inclass presentations: 200 points total
- Total: 400 points

**GOALS OF THE COURSE:**
Since mathematics is a useful tool in everyday life and in many academic fields,
it is taught in elementary courses in terms of its applications.
Indeed most people think that mathematics is synonymous with applications
of mathematics, such as balancing a check book, finding the area of a field,
computing the stress on a beam, or determining the orbit of a satellite.
These are examples of using mathematics to solve
problems in other disciplines. They are not problems in mathematics itself.
The view that applied mathematics is all of mathematics is no more a
true picture of mathematics than it would be to think that commercial
art is all of art. What, then, is mathematics? One of the goals of
this course is to provide you with a clearer answer to this question.

A real mathematician is not just a user of mathematics, but is also
an *inventor of mathematics*. This course will give you a
glimpse of the work of a mathematician.
You will invent your own proofs to the many theorems stated in the notes.
You will discover your own counter-examples and solutions to
the various questions and problems.
This course requires you to * actively participate*
at a level beyond that assumed in an ordinary course.
In a standard course, the homework done outside of class
reinforces the material presented in lectures or in the text.
In this class, however, student generated proofs form the foundation
of the course.
Just as a music student could not possibly learn how to play
an instrument by listening to someone else play,
you cannot become a mathematician merely by seeing another person's proofs.
Think of this course as like driver's ed in high school, where you were
expected to spend your share of time behind the wheel of the training cars.
To draw another analogy, this time from sports,
the only way to learn to catch a baseball is to put on a glove and play catch.
In this course we play ball in a game called number theory, using the rules of
logic.

**RULES OF THE COURSE:**
There are five golden rules:

- 1.
*Members of the class will present their own proofs of the theorems and solutions of the questions and exercises at the board.*It is the job of the rest of the class to spot any errors in the proof being presented. As your instructor---and therefore a member of the class---I too can give my own proofs. In general, however, I hope to keep the time I spend in front of the class to a minimum. The more you do on your own, the more you will master this subject. - 2.
*Other textbooks are forbidden.*Proofs to most of the theorems in these notes can be found in standard textbooks on number theory. Please do not give into the temptation to look at them. It will spoil the problem for you and other students in the class. - 3.
*No outside help.*Do not seek answers from faculty members, advanced graduate students, or other math majors not enrolled in the class. You may talk to me outside of class. Show me what you've done and I might be able to throw you a lifeline. - 4.
*Minimum collaboration.*Discussions during class about how to prove the theorems may naturally continue when class is over. I do not mind*limited*discussions outside of class, but I do want you to tackle these problems individually, not as teams. - 5.
*You can leave during a presentation.*Suppose you have been working for several days on a proof, when another student is ready to give a solution in front of the class. If you wish to keep working on the problem, you may elect to temporarily leave the classroom while the other proof is presented.

**HANDOUTS FROM THE TEXT:**
The text

- Math 303 Introduction to Number Theory: Lecture Notes
- by Richard Blecksmith

NOTE: Each module is in PDF format, so you will need Adobe's Acrobat Reader which is a free and useful download. Click on the above link to get the latest version of the Acrobat Reader.

- Challenge Problems (1-5 handed out in class)
- List of Primes to 10,000
- Module 1. Divisibility and Congruences
- Module 2. An Elementary Approach to Primes
- Module 3. The Fundamental Theorem of Arithmetic
- Module 4. Mediants, Contiguous Numbers, and Farey's Sequence
- Module 5. Fibonacci Numbers
- Module 6. Partitions
- Module 7. Pythagorean Triples

**STUDY GUIDES:**

EXAM 1. (Oct 29)

EXAM 2. (Dec 5)

**THANKSGIVING BREAK:**
Thanksgiving Break is from November 26 through November 30.

**FINAL EXAM:**
Your final exam will be given in class at the end of the semester,
probably on Wednesday, Dec 3, 2012. It is a 100 point exam, covering
the material in the second half of the semester.
The Final Exam is scheduled for 4:00 - 5:50 p.m., Wednesday, Dec 10, 2014.
We will meet during this time to discuss the results of this second test.

** About the text:**
In a way this text will be one of the easiest math books you've
ever read. Many of the sections are written in dialogue form,
with a minimum of formulas and a lot of space spent on simple
explanations. Let's face it, most math students don't
read the textbook, except for the exercises and possibly the
examples - to be read only when you get stuck on the exercises.
Hopefully this book is different. Try reading it and see.

In another way this text will be one of the most challenging
math books you might have ever been exposed to. It expects you
to **think** and thinking - especially in mathematics - is
hard work. But it can also be fun. That's why the cross-word puzzle
is often found near the comics in the newspaper.
Thinking isn't so bad if you believe that what you're
thinking about is worth the effort.
The difference between mathematicians and the rest of the world
is that mathematicians actually **like** thinking about math.
It may surprise you to discover that mathematics is not as
boring and humdrum as you might believe. Many people who dislike
the subject can still get interested in topics such as
infinity, chaos theory, space-filling curves, and logical
paradoxes. In this course you will discover that not
all infinities are the same; that you can have curves with
a finite area, but an infinite perimeter; that you can
reorder the points on the number line so that they do not
take up any length.

One warning: Not all the questions posed in the text are answered for you; not all the theorems and facts are proved for you. You will have to work many of them out yourself - possibly with the help of your instructor or classmates. This is not your typical "how-to" course. Your job, as a student, should not be rote utilization of formulas, but rather reasoning, thinking, and possibly discovering.

**Advice about the course:**
Perhaps the single most important factor in your success
in this course is your ** study habits **.
Think of learning math as "working out" in the gym.
Study at least 3 times per week; do not wait until the day before the exam.
Learn mathematics like you would learn a language.
Work on the concepts until they make sense.
Don't just memorize facts and then forget them a few weeks later.
Master as many homework problems as you can - beyond just getting a
correct answer.
** Always come to class! **
While you're there, listen, think, and ask questions.

Last update: Sept 20, 2012