n Number Theory 189-346 / 377B Page, McGill, Spring 2013

189-346 / 377B: Number Theory

Course Projects:
The course projects should be handed in before Thursday, April 25 (midnight).
You are strongly encouraged to submit your project electronically, as a pdf file, to darmon@math.mcgill.ca. Please use only pdf format. If you are unable to submit an electronic version, I will also accept hard copies in my mailbox (before Thursday at midnight) but electronic submission is strongly preferred.

On the request of some of the students in the class, I plan to post the student projects on this web site so you can all have access to them (as well as the rest of the world...) If you object to your work being publicly posted, just let me know when you submit your project and I will not post it.

Here are some of the projects:

Colleen Alkalay-Houlihan. Sophie German and special cases of Fermat's Last Theorem.

Mathilde Gerbelli-Gauthier. Le Théorème des unités.

Azeem Hussain. The Agrawal-Kayal-Saxena Primality Test

Benjamin Inch. The analytic development of the p-adic numbers

Akash Jaggia. On Fermat's method of infinite descent

Fang Xi Lin. Shor's algorithm and the quantum Fourier transform

Santiago Paiva. Elliptic curve cryptosystems

William Wright. Dirichlet's theorem on primes in arithmetic progressions

Professor: Henri Darmon
Classes: MWF 10:35-11:25, BH 920.

Office hours:
Henri Darmon: MW 2:00-3:00, or by appointment, in room 1111.

You may also avail yourself of the services of the Math Help Desk, which is open Monday-Friday from 12:00 to 5:00 PM, in BH 911.

Recommended Texts:

Andrew Granville, Introductions to Gauss's Number Theory, together with the appendices.

This will be the primary text for many of the topics covered in the syllabus. It can be downloaded freely from this web page. Note that this textbook is still in progress and is not yet complete. In fact, it is being tried out in the classroom for the first time! Any feedback that you can provide me in your role as guinea pigs (typos, grammatical errors, obscurities of exposition, etc.) will be forwarded to the author and will be most appreciated.
To quote the author himself, "This is the first draft of an introductory book of number theory, where the focus is on trying to explain proofs so that they are easily accessible. I need your help, especially in knowing which parts are not easy to understand. Please email me with your comments, good and bad, on this (and any other suggestions) to andgranville@gmail.com. Thanks, Andrew Granville"

William J. LeVeque, Fundamentals of Number Theory, Dover Books.
I have chosen this text because it gives a good overview of the standard topics in the subject with a historical slant which I like. It is available in a cheap Dover edition, and acquiring it will not set you back by more than the price of three Lattes at Second Cup. Very highly recommended.

Winfried Scharlau and Hans Opolka, From Fermat to Minkowski: Lectures on the Theory of Numbers and its historical development.
This textbook gives a historically motivated account of the subject at a somewhat more advanced level than the other two books. It is rather pricey and is only proposed as an optional text for those (particularly the students in 377) wishing to go a bit beyond the material covered in class.

Syllabus: This course will cover the standard syllabus for an introductory undergraduate course in number theory. The content and pace will be challenging: emphasis will be placed on rigorous proofs, and on developping mathematical maturity and problem-solving skills.

Grading Scheme :

346: 20% Bi-weekly assignments , 40% midterm, 40% final exam.

377: 20% Bi-weekly assignments , 20% term project, 20% midterm, 40% final exam.

The corrections to the midterm are now available.

Alternate schemes: If you do better on the final than on the midterm, the midterm will be discarded and the final will count for the balance the final grade (80% for students in 346, and 60% for those in 377). The component of the grade based on the assignments and term project can not be made up for by a strong performance in the final exam.

Computers:

Computation and experimentation are an important facet of Number Theory, a tradition that does back at least to Gauss who was a prodigious calculator. Because of this, Number Theory is the branch of pure mathematics that is perhaps the closest to physics. (This may seem surprising in light of Number Theory's reputation as the purest part of pure mathematics, well removed from the "real world".)

Unlike physics where experiments often rely on costly apparatus that can only be carried out in well-endowed laboratories, the requirements for experimentation in number theory are modest: a personal computer running a symbolic algebra package is all that you will need. A number of questions in the assignments will rely on calculations on such a symbolic algebra system. Pari/GP, which is freely available on the web, is the system I recommend. (But you are free to use an equivalent system, like Maple, Mathematica or Magma if you prefer.)

Before writing Assignment 1, you should download Pari onto your computer. You might want to seek help from a classmate if you have trouble in doing this. I am told that the operation is relatively simple and painless if you work in a Windows environment, but if you are part of the swelling crowds of Mac afficionados, it can be a bit more compicated. Xiangyu Wu has been through the installation successfully, and has kindly sent me these instructions:

1) Download and install XCode by going to
http://connect.apple.com/.
You need to register for apple developer, it is free. Once registered, login to the developer portal and search for XCode.

2) Download and install Mac Ports:
http://www.macports.org/install.php.

3) Download the file Pari from:
http://pari.math.u-bordeaux.fr/download.html

4) Download and install Command Line Developer Tools from the same site as XCode.

5) Launch Terminal and type "xcodebuild -license" from the command line prompt.

6) Then, type "ports install pari" from the Terminal command line prompt.

Manual. Valery Lacoste has kindly pointed out the availability of the following short Pari tutorial which she found on-line and which some of you might find helpful.

The usual disclaimers:

Academic Integrity: McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see Academic Integrity for more information).

L'honneteté académique. L'université McGill attache une haute importance à l'honneteté académique. Il incombe par conséquent à tous les étudiants de comprendre ce que l'on entend par tricherie, plagiat et autres infractions académiques, ainsi que les conséquences que peuvent avoir de telles actions, selon le Code de conduite de l'étudiant et des procédures disciplinaires (pour de plus amples renseignements, veuillez consulter le site Academic Integrity.)

Submitting work in either of the Official Languages. In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

Syllabus and Grade Calculation. In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.