Combinatorial Number Theory
Spring 2004
Mathematics V3021 section 001
M, W 10:35-11:50 am
520 Mathematics Building
http://www.math.columbia.edu/~glass/spring04/
Darren Glass
Office: 516 Mathematics Building
Office Phone: (212) 854-5135
glass@math.columbia.edu
The best way to get in touch with me is by email. I am an email addict.
Announcements
Check this part of the website regularly for announcements.
- 4/26/04 - We wil have the final presentations on May 5th at 11am in
our normal room. I anticipate it running about 2 hours. I will provide
food and beverage. Let me know ASAP if this is a problem for you.
- 4/20/04 - I will have extra office hours tomorrow from 12-3 if people
want to drop by to discuss their projects (or anything else)
- 4/4/4 - As we enter the "Continued Fractions" section of the course, I
would continue to recommend the Kumanduri textbook, as well as Chapter Ten
of the classic Hardy and Wright's Introduction to the Theory of
Numbers. If you really want an online reference, check out William
Stein's lecture notes. Lectures 17-20.
- 4/1/04 - I guess it doesn't do much good for me to write the official
project description if I am not going to link it to
the website. Oops.
- 3/30/04 - This looks like a good opportunity
for any of you who have some spare time this May.
- 3/27/04 - Several of you have asked for a good source on Quadratic
Forms other than the Kumanduri book, which I have been following pretty
faithfully. John Conway has an excellent book entitled The (Sensual)
Quadratic Form which covers similar material (and more material, which
might make good final projects...)
- 3/21/04 - Presentations begin on Wednesday. You should bring me back
a written copy of the project as well -- either the version you already
handed back or a new version taking into account any suggestions I made.
- 3/21/04 - Homework 5 is posted below. It will be due on Monday, April
5th. If this is a problem due to Passover talk to me and we can work
something out.
- 3/3/04 - I will be out of town so there are no office hours next
monday (class will meet as normal). To make it up to you I am having an
extra office hour tomorrow from 2:30-3:30 and one right now!
- 3/2/04 - There was a problem with question 3 of homework four as it
was written (don't blame me -- blame the book I stole the problem from!)
in that the form was indefinite. So I changed one of the coefficients of
the form. The version that is up now is corrected.
- 2/26/04 - I've updated the description of the midsemester project
below with all the new deadlines. I have also posted HW4 below.
- 2/22/04 - I've added more ideas for the
end-of-semester project. It's never too early to start thinking about
it. (And you definitely need to start thinking about the midsemester
project if you have not done so!)
- 2/9/04 - This is an interesting article I found
about the limits of how far the techniques I talked about today in class can be
pushed and just what information is known about them.
- 2/9/04 - I have posted below the new homework as well as a more detailed
description of the midsemester projects. As always, you should look these over
sooner rather than later.
- 1/28/04 - I will have an office hour tomorrow (Thursday) from 2:30-4.
- 1/26/04 - due to popular demand, homeworks will be due on Wednesdays.
So the first one (posted below) is actually due on 2/4.
Course Information
Content
Number Theory is, in my rather biased opinion, the most beautiful area of
mathematics. It is one of the oldest area of mathematics, and in recent
years a great number of applications to cryptography (and other things)
have been found.
This semester we will be starting off by discussing the prime number
theorem and Dirichlet's theorem on the distribution of prime numbers and
we will see where that takes us.
Warning: This is the second semester of number theory. I assume
that if you did not take my course last semester then you have taken the
first semester in a previous year or elsewhere. If you want to know what
I expect you to have seen before, you should look here.
A list of what I have done each day can be found at this website
Textbook
There is no official textbook for this course, as I will be pulling from a
variety of places. At times I will refer to Silverman, as most of you
already own that book.
We will start by from the notes Not Always Buried Deep by Paul
Pollack which are available on the web here.
I often find it helpful to look at multiple books to get a variety of
viewpoints on the same material. If you wish to do this, some of the
books I reccomend are:
- A Friendly Introduction to Number Theory (Second Edition) -
Joseph Silverman
- Elementary Number Theory - Underwood Dudley
- An introduction to Number Theory - PD Schumer
- Number Theory with Computer Applications - Ramanujachary
Kumanduri & Christina Romero
- Making, Breaking Codes - Paul Garrett
- Lecture
Notes on Cryptography - Shafi Goldwasser & Mihir Bellare
Office Hours and Other Help
Number Theory is not an easy subject, and it is almost certain that you
will need help during the semester. Please do not hesitate to take
advantage of the various forms of help that we provide. The Columbia Help
Room is Mathematics 406. You may go there anytime it is open to have
your questions answered.
Other office hours: I will be in Mathematics 516 on Mondays from 1:30-3.
Drop by sometime to chat.
Grading
50% - Homework
10% - Attendance and Participation
40% - Project/Presentations
Homework
Homework will be assigned roughly once a week. You are encouraged to
discuss homework with each other, but you must turn in your own work. If
you aren't sure exactly where the boundary lies, please ask me rather than
make assumptions. Late homework will be penalized unless prior permission
has been obtained from the professor. All homework will be graded not
only on correctness, but also on how well you communicate and explain your
answers. Mathematics is a process, not a final answer, and your work
should reflect that.
For many of you, this will be the second course where you are writing
proofs. Learning to do this well is a slow and difficult process. Some
hints you may find helpful are here.
- Homework Set One (due Feb. 4th) - DVI PDF
- Homework Set Two (due Feb. 18th) - DVI PDF
- Homework Set Three (due Feb. 25th) - DVI PDF
- Homework Set Four (due Mar. 10th) - DVI PDF
- Homework Set Five (due Apr. 5th) - DVI PDF
- Homework Set Six (due Apr. 19th) - DVI PDF
- Homework Set Seven (due Apr. 28th) - DVI PDF
Midsemester Projects
Sometime before spring break, you need to solve one of the problems in the
book Problems in
Elementary Number Theory. You should write up your solution and then
present it to the class in 5 or 10 minutes. I recommend that you turn in
the written version before you present it to the class. Only one student
can do each problem, so if you want to claim a problme you should email me
to tell me which one. A list of the claimed problems is here. You will be graded both on correctness and on
presentation.
Official Project Description
Final Projects
For this course you will have to write a final paper and present it to the
class. While these papers can have a historical component they must also
have significant mathematics in them.
The goal of the written projects is to investigate mathematical ideas in
greater depth, to understand the connections between different topics, and
to develop skill in writing and speaking about mathematics. Here are all the details.
You are encouraged to come up with your own topics, but here are some ideas to get you brainstorming. I will add
to this list periodically. You will have to submit a proposal to me right after
spring break.
Other Resources
Norm Thompson