I plan to cover Chapters 1-9 of Rosen's text.
Chapters 1 and 2 are very elementary. They introduce divisibility, primes and the Euclidean algorithm. The main theorem in those chapters is the Fundamental Theorem of Arithmetic which says that every integer greater than 1 can be factored uniquely as a product of primes. This fact is taught (without proof) in elementary school.
Chapters 3 and 4 are about modular arithmetic and its applications. Probably the main theorem here is the Chinese Remainder theorem which can be used to do problems in usual arithmetic (like addition, subtraction or multiplication) faster by breaking them up into several easier modular arithmetic problems.
Chapters 5 and 6 are about a few of the famous old theorems in elementary number theory starting with Wilson's theorem and Fermat's little theorem and ending with Euler's generalization of Fermat's little theorem. Fermat's little theorem is easy to state. It says that, if p is prime and n is any integer, then p always divides np-n. It is not really that difficult to prove either, but it has turned out to be very important both in number theory and recently in applications of number theory to cryptography.
Owing to time pressure, we will probably do very little of Chapter 7. Chapters 8 and 9 are on primitive roots and Gauss's Law of Quadratic Reciprocity. This last result is really the bridge between classical and modern number theory. I have read that it was Gauss's favorite of his many theorems.
There will be two midterm held in class. The dates are as follows.
| Midterm 1 | 11 October 2012 |
|---|---|
| Midterm 2 | 15 November 2012 |
Midterm 2 will cover the material in the homework up to and including Homework 11. There will be special emphasis on the material that has been covered since the first midterm.
Course grades will be computed as follows.
| Homework | 20% |
|---|---|
| Midterms | 40% |
| Final | 40% |
Homework will be assigned weekly and due on Thursdays in class. It will be assigned on the homework web page linked here and at the top of this syllabus web page. Late homework will not be accepted but the lowest homework grade will be dropped.
I want every student to feel free to questions in class. Without student questions, even ones that might sometimes seem silly, lectures can become very dry and monotonous. So questions and comments are more than welcome!
On the other hand, please be courteous to other students by paying attention to the lecture and not carrying on conversations with each other. These kind of private conversations can be distracting both to me and everyone else.
Please make sure that your cell phone does not ring during the lecture, and please do not use headphones. Please do not use laptops for anything not related to the class, and, if you do bring a laptop, please sit in the back so that your screen is not distracting to others.
Applications are due Thursday for the Maryland Directed Reading Program. Students can learn more and apply online at http://drp.math.umd.edu .
If you want to get an idea of what sorts of things you might do in the Maryland Directed Reading Program, the talks from this past summer are Tuesday and Wednesday evening, from 5 to 7, in the Math Lounge across from Math 3206.