**Combinatorial Number Theory**

Mathematics V3021: Spring 2002

**
Detailed information on homework and exams**

** Course Description:** The main topic of this class is
the beautiful and accessible subject of Number Theory.
We will focus on the investigation of points on curves,
a topic which has fascinated people for centuries.
We will learn about some applications of Number Theory to coding theory and
cryptography.
Some topics are: Diophantine equations and approximation, quadratic fields,
continued fractions, finite fields, projective space, elliptic curves,
and coding theory.

**Course Philosophy:** One goal for this course is for everyone to
be actively involved and to explore material through
experimentation. Another is to develop skill at writing proofs.
The prerequisites are V3020 or W4041 or the equivalent.
As usual, the amount you learn will depend mostly on how hard
you work on the material.

**Examinations:**
There will be a midterm given
in class on Wednesday, March 13th.
There may be final presentations during the final examination
period Monday, May 13th, 9 am- 12 noon.
*
There are no makeups for missed exams,
regardless of the reason for absence*.
*You must take the
final examination at this time scheduled by the university; no final exams
will be given earlier.*
If you have two final examinations scheduled at the same time, it is the
responsibility of the other department to provide an alternate exam.
Examinations will not be rescheduled because of travel arrangements. It
is your responsibility to schedule travel appropriately.
*
During the presentations (4/29, 5/1, 5/6, 5/13), attendance is mandatory
*.

**Grading: **The course grades will be computed as follows.

40% Homework and Class Participation;
25% Midterm; 35% Final Project and Presentation.

**Text:**
1)
A Friendly Introduction to Number Theory (Second Edition),
Joseph Silverman
Publisher Prentice Hall.
If you do not already have this book, you do not need to buy it (about $70).
It is possible to borrow it for the semester.

2) Course packet. This contains selections from:
Davenport: A Higher Arithmetic, Artin: Algebra, and Silverman/Tate:
Rational Points on Elliptic Curves. This course packet (about $20)
can be purchased through the Columbia math department.

3) Codes and Curves by Judy Walker.
This short book ($12) can be purchased on discount
through the Columbia math department.

All of the above textbooks and course packets are on reserve in the
library. Also on reserve are books by Ireland/Rosen and Kumanduri/Romero
which might be useful for the final projects. Please do not take the reserve
books out of the library.

**Homework:**
Homework is the most important part of this class.
It should demonstrate your knowledge of the material,
your investigation of open ended questions, and your skill at
writing proofs.
Homework is due every Friday at 10:30.
Homework must be neat, legible, and stapled in order to receive credit.
You may hand in the homework in class, under my door, or to my mailbox.
Late homework will be given a score of no more than 50%.
The lowest homework score will be dropped. Both independent and group work
is encouraged.

**Help:** Help is available if you have trouble with homework
or lecture material. If you are unable to attend my office hours,
go to the
Mathematics
Help Room (406 Mathematics). This room
is always staffed by at least one mathematics professor or graduate student
Monday through Friday during business hours. You may drop by whenever the
Help Room is open; no appointment is necessary.

**Method of Study:**
Your work on every topic should include the following:

1. Reading the book.
2. Contributing to class.
3. Searching for data.
4. Discussing material with others.
5. Creating conjectures.
6. Proving statements.
7. Asking new questions.