Northern Illinois University

DuSable Hall 302

Monday, Wednesday 6 - 7:15

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 364
- email: richard@math.niu.edu
- Phone: 753-6762
- Office Hours: Mon, Wed 4-6

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

- Module 1. Computers, Codes, and Binary Numbers
- Binary Numbers
- Error Correcting Codes
- Pails of Water
- Clock Arithmetic
- Secret Codes

- Module 2. An Elementary Approach to Primes
- The Fundamental Theorem of Arithmetic
- GCD's and Katie's Theorem
- Excursion into Word Problems
- The Distribution of Prime

- Module 3. Exploring Fractions
- The Mediant and Victoria's Theorem
- Contiguous Fractions
- The Farey Sequence

- Module 4. Mathematical Heresy
- Square root of 2 and Fermat's Method of Descent
- To Infinity and Beyond
- Counting Fractions
- Irrationals - the Dark Matter of the Real Number Line
- Third Deadly Sin: square root of -1

- Module 5. Partitions

**WITHDRAWAL:**
The last day for undergraduates to withdraw from a full-session course
is Friday, October 21.

**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

**ABSCENCES:**
Everyone is allowed to miss one class without penalty.
After that, every unexcused absence will result in 10 points deducted
from you total 400 points.

**GOALS OF THE COURSE:**
Scattered throughout these notes are 18 activities, consisting of
games such as "Nim" and "Krypto," brainteasers such as "Mental Math"
and "Hat Problems," and routine calculations needed to find
"Ruth-Aaron Numbers" or to solve the "Postage Stamp Problem."
Typically I split a class into groups of 3 or 4 students,
asking each group to work on that day's activity, by talking among
themselves, brainstorming ideas, and dividing up routine calculations
(busy work). Working alone, however, a motivated student, with
perserverance and energy, should be able to complete most of these tasks.

Let's face it, many college mathematics texts go unread, except for homework and selected examples, because they confuse and frustrate their readers. I wanted this text to be something a student could actually read and understand. The sections are split into small paragraph--sized subsections, each containing a predigested idea, ready for consumption (like feeding a baby bird a worm.) Sections do not end with a list of dozens or more boring and routine exercises. Despite this dearth of repetitive exercises, there should be plenty of things for you to think about. Ideas, strange examples, questions, analogies, and some humor (mostly bad) abound on these pages.

Some topics require a certain level of mathematical rigor, a good thing
for students to be exposed to. Mathematics differs
from all other subjects in that we can * prove * what we claim
to be true. If you add two odd numbers, the sum will be even. Every time.
And there is a proof that explains why this is so.
I felt it important to give precise proofs of such important material
as the Fundamental Theorem of Arithmetic and the Infinitude of Primes.
Many proofs, with hints, are left to the reader.

As the title of the book suggests, our goal is to go "beyond the standards" set for the K-12 public school curriculum. You will be introduced to topics such as encryption and primality--testing, not normally seen in an elementary course. To be sure, you won't become an expert in cryptology by mastering the simple 31--code in Module I, any more that you will be ready for a career in Stand--up by decoding the Jokes in Activity 6. But you should get a feeling for how number theory plays a role in modern technology and secure communications.

I hope that you will like thinking about some of the topics presented here. After all, "thinking" is a good thing and "liking math" is something you will pass on to your own students.

- Beyond the Standards: Exploring the World of Numbers
- 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.

- 1. Multiplying with your Fingers (1-5 handed out in class)
- 2. Nim
- 3. Krypto
- 4. The 3 Pails Problem
- 5. The Postage Stamp Problem
- 6. Jokes
- 7. The Sieve of Eratosthenes
- 8. The Ruth - Aaron Numbers
- 9. Math Magic Trick
- 10. Guess the Number
- 11. Primality Testing
- 12. Fibonacci Numbers
- 13. Pythagorean Triples
- 14. Fooled Again
- 15. Factoring Primes
- 16. The ABC Problem

- Introduction
- Section 1. Binary Numbers
- Section 2. Error Correcting Codes
- Section 3. Pails of Water
- Section 4. Clock Arithmetic
- Section 5. Secret Codes

- Introduction
- Section 1. The Fundamental Theorem of Arithmetic
- Section 2. GCD's and Katie's Theorem
- Section 2(b). Reducing Fractions
- Section 3. Exccursion into Word Problems
- Section 4. The Distribution of Primes

- Introduction
- Section 1. The Mediant and Victoria's Theorem
- Section 2. Contiguous Fractions
- Section 3. The Farey Sequence

- Section 1. Square root of 2 and Fermat's Descent Method
- Section 2. To Infinity and Beyond
- Section 3. Counting Fractions
- Section 4. The Irrationals - The Dark Matter of the Real Number Line
- Section 5. Third Deadly Sin: the square root of -1

**STUDY GUIDES:**

EXAM 1. Monday, Oct 21

EXAM 2. Monday, Dec 9 (6 - 7:50 pm)

Last update: Sept 20, 2012