189-346 / 377B: Number Theory

Professor: Henri Darmon
Classes: MWF 10:35-11:25, BH 1B39.

Office hours:
Henri Darmon: M 2:00-3:00 and F 11:30-12:30, 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.

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.

Here is an old practice midterm.
Here is the actual midterm 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.

The final exam will be on April 19, at 9:00 AM.
Because of the conflict with the Passover holiday, there is the possibility of writing a special exam on April 29. Students wishing to avail themselves of this possibility must fill in a conflict form (which can be found at this link) that is signed by their Rabbi, attesting to the fact they cannot be present on the day of the exam due to religious obligations.

The final grades need to be handed in one week after the final exam, therefore I'd like to ask you to hand in your course projects on Monday, April 25. (You can slide them under the dor of my office, or place them in the 10th floor mailbox.)

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.

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.