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**Trinity College **

**Math 253-01 Number Theory **

** TR 1:30-2:45, Nutt MECC 232 **

**Text: Elementary Number Theory, 7 th edition, by Burton **

**Instructor: Professor Nancy J. Wyshinski **

**Office Hours: M12-1, T11-12, W11-1, R3-4, others by appointment, **

**Nutt MECC 241, x2221 email: nancy.wyshinski@trincoll.edu**

Number theory is the study of the natural numbers (or positive integers)

1, 2, 3, 4, 5, … and the relationships between different types of numbers. For example, odd

1,3,5,7,… prime

2,3,5,7,… cube perfect

1,8,27,64,…

6,28,496,…

Fibonacci

1,1,2,3,5,8,…

Some of these types are already known to you, and some may be new. We will encounter all of these types and more in this course. We will discover some great and unexpected relationships between types of numbers and prove they are true. In addition, we will consider some number theoretic problems, some we will prove and some which we cannot (either because we do not have the tools, or because no one can)! Examples –

Are there infinity many primes?

Are there infinity many primes that have a remainder of 1 when divided by 4?

Are there infinity many primes that have a remainder of 3 when divided by 4?

Can the sum of two squares be a square?

Can the sum of two cubes be a cube? What about fourth powers? Fifith?

Twin primes – what are they and are there infinity many of them?

Perfect numbers – are they always even?

This will be the best math class you have ever taken!!!

**Important College Dates **

Tuesday, January 29 – Add/Drop Period ends

Friday, February 15 – Last day to withdraw from a course

April 8 – 12 – Advising Week

April 15 – 22 – Advance Registration for Fall, 2013

Wednesday, May 1 – Last day of classes

May 2 – 5 – Review Period

May 6 – 10 – Final Exams Week

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1/22

**Tentative Lecture Schedule **

**Tuesday **

Ch 1 – Preliminaries 1/24

**Thursday **

Ch 1 – Preliminaries

1/29 Ch 2 – Divisibility Theory 1/31 Ch 2 – Divisibility Theory

2/5 Ch 2 – Divisibility Theory 2/7 Ch 3 – Primes and their

Distribution

**2/12 2/14 Test #1 **

2/19

Ch 3 – Primes and their

Distribution

Ch 4 – The Theory of

Congruences

2/21 Ch 4 – The Theory of

Congruences

2/26 Ch 4 – The Theory of

Congruences

2/28 Ch 4 – The Theory of

Congruences

3/5 Ch 5 – Fermat’s Theorem 3/7 Ch 5 – Fermat’s Theorem

**3/12 Ch 6 – Number Theoretic 3/14 Test #2 **

Functions

3/19 Spring 3/21 Break

3/26

4/2

Ch 6 – Number Theoretic

Functions

Ch 7 – Euler’s Generalization of Fermat’s Theorem

3/28 Ch 7 – Euler’s Generalization of Fermat’s Theorem

4/4 Ch 8 – Primitive Roots and

Indices

4/11 Ch 9 – Quadratics Reciprocity 4/9 Ch 8 – Primitive Roots and

Indices

4/16 Ch 9 – Quadratics Reciprocity 4/18 Ch 13 – Representation of

Integers as Sums of Squares

4/23 Ch 13 – Representation of

Integers as Sums of Squares

**4/25 Test #3 **

4/30 Ch 14 – Fibonacci Numbers

**Final Exam **

**Friday, May 10 **

**9-11am**

**Grading **

Homework: Graded assignments will be given at weekly. You are encouraged to work with others on homework. Always include the names of other students that you have collaborated with but submitted work must be your own. The final homework score will be based on 190 points. No late homework will be accepted.

Tests: We will have three exams scheduled for the semester, each worth 100 points.

Attendance: I will take attendance daily. You may miss three times without penalty; after that, your final grade will decrease by .5% for each absence. Please do not let this happen to you!

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Final Exam: Our final exam is scheduled for Friday, May 10, 9-11am. The final will be worth 100 points.

Math Journey Points: Math Journey Points are 10 special points earned by doing mathematics outside of the classroom. You must obtain the points before the last day of our class. Points may not be obtained during reading days or finals week. You may also earn an additional 10 points as extra credit.

Attending a mathematics talk and writing a review of it (4 points) (up to 8 points)

I will announce talks in class as well as post them on Moodle.

Attending the weekly Math Department Tea (2 points) (up to 6 points)

Having lunch at the Math Table at Mather (2 points) (up to 6 points)

Reading a mathematical book and writing a review of it (4-5 points) (up to 10 points) You need to read the book during this semester and cannot simply write a review of a book you read in your past. I have many books to choose from, such as ….

The Square Root of Murder , by Ada Madison, The Probability of Murder , by Ada

Madison, A Function of Murder, by Ada Madison

The 5 Elements of Effective Thinking, by Burger and Starbird

The Number Devil, by Hans Magnus Enzensberger

The Man Who Counted, b y Malba Tahan

Flatland, by Edwin A. Abbott

My list is by no means exhaustive which means you may suggest a book to read

(and when approved, it will then be included on our list). I will keep an ongoing list on Moodle.

Watching a math movie and writing a review of it (4 points) (up to 8 points)

*Again, I have many dvd’s for you to choose from, and our library is a terrific source as well. Some examples from my collection are: A Beautiful Mind, Julia *

*Robinson, Fractals, The Story of 1, π, Good Will Hunting, Flatland, The Great π/e *

*Debate, The Story of Math. *

**Also, from our Trinity Library - Films on Demand**

Description: Web-based digital video delivery platform that allows viewing of thousands of streaming videos from Films Media Group anytime. Available for in-class use and remote viewing.

Coverage: Varies

Notes: Works best with Firefox or Internet Explorer. Use the 'email this video' feature to send a film link, not the browser link. When linking to a direct title, use ONLY the authenticated Title URL or Segment URL that is located at the bottom of the video display.

Search Tips: Register on campus in order to take advantage of all features. Once registered, you may log in from anywhere.

Bookmark Link: http://library.trincoll.edu/databases/filmsondemand

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**Final Semester Grades **

There are a total of 600 possible points (3 x 100 + 190 + 10+ 100 = 600). Your final grade will be based on the following percentage table:

**90%-100% **

**80-89% **

**70-79% **

**60-69% **

**At least **

**A- **

**B- **

**C- **

**D- **

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