Divisibility, prime numbers, linear congruences, multiplicative functions, cryptology, primitive roots, and quadratic residues.This is vast underselling of the subject of this course: Number Theory is one of the oldest of the "true" mathematical disciplines (= areas of mathematical investigation done in a way we would recognize today) ... perhaps the 1.9th oldest. It has astonishingly simple yet astonishingly beautiful results. It has rich structure and depth which works smoothly together like an intricate machine, and yet has consequences for statements simple enough that an elementary school student could understand.
One delightfully ironic aspect of Number Theory is that it was thought of for a couple thousand years as the purest of pure mathematics, and it should be therefore be innocent of good and bad application. The great English mathematician G.H. Hardy wrote in his book A Mathematician's Apology (published in 1940):
No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.Hardy would probably be terribly disappointed to know that Number Theory underlies the great majority of techniques of ensuring security and privacy on the Internet, to the point where, for example, the US National Security Agency is the world's largest employer of Ph.D. mathematicians, many of them number theorists.
During the first, foundational part of the course, we will cover a portion of the basics which corresponds roughly to chapters 1–15 & 20–23 of our textbook. This should take us approximately 3/4 of the term. A sightly more indicative list (than the catalog description) of topics in this first part would include
- linear and quadratic Diophantine equations ... hints of higher degrees
- divisibility, the greatest common divisor
- primes
- the Fundamental Theorem of Arithmetic
- congruences
- the Chinese Remainder Theorem
- Wilson's Theorem and Fermat's Little Theorem
- arithmetic functions, Euler's Φ function
- primitive roots
- quadratic residues and quadratic reciprocity
- cryptology
- Diophantine approximation
- continued fractions
- p-adic numbers
Class organization: The textbook we are using is, as it says, quite friendly. [Almost too friendly.] We will center our activity in the course around the book, covering a chapter approximately every two classes, in the following way:
- Keep an eye on the course schedule page, at least as frequently as we have class.
- Read the chapter we will be discussing before the class in which we will discuss it (and probably again after the first class discussion of that material).
- Every student must submit to the course web site thoughts and/or questions that have come up during their reading and thinking about the chapter, at least an hour before each class. [We'll call these your T&Qs.]
- The course schedule page will also name two students for each chapter as student chapter leaders [SCLs]who will be responsible for taking and writing up class notes. These students should work together and make absolutely sure they have clear, complete, formal versions of all the definitions, statements (lemmata, propositions, theorems, corollaries, etc.), and proofs.
- The SCLs will get their material into our electronic Class Collaborative Textbook [CCT]. This text will be available at all times to all students, and should provide a great help by filling in all the unfriendly details the paper textbook is missing.
- Along with each class discussion of a chapter, we will be discussing several of the chapter's problems. One such should also go in the CCT, as part of the SCLs' write-up.
- Students who are not SCLs for a given chapter will have 2–4 homework problems to do and to write up from each chapter. These must be turned in at the next class after we are done with that chapter.
- You may turn in the HW on paper or electronically. If you use paper, you will be given 1 point of green points extra credit for every page you reuse: take a page from some other source (handout you are finished with from another class, draft page of a paper you are writing, whatever), put a big X through the written part and do your math work on the (previously blank) back of the paper. If you do electronic HW submission, you will automatically get 5 green points for every set.
Revision of work on homework, CCT work, and tests: A great learning opportunity is often missed by students who get back a piece of work graded by their instructor and simply shrug their shoulders and move on. In fact, painful though it may be, looking over the mistakes on those returned papers is often the best way to figure out exactly where you tend to make mistakes. If you correct that work, taking the time to make sure you really understand completely what was missing or incorrect, you will often truly master the technique in question, and never again make any similar mistake.
In order to encourage students to go through this learning experience, I will allow students to hand in revised solutions to all homeworks, CCT sections, and midterms. There will be an expectation of slightly higher quality of exposition (more clear and complete explanations, all details shown, all theorems or results that you use carefully cited, etc.), but you will be able to earn a percentage of the points you originally lost, so long as you hand in the revised work at the very next class meeting. The percentage you can earn back is given in the "revision %" column of the table in the Grades section, below.
Exams: We will have two midterm exams on dates to be determined (and announced at least a week in advance). These may have a take-home portion in addition to the in-class part. Our final exam is scheduled for Friday, May 4th from 10:30am-12:50pm in our usual classroom.
Grades: On exam days or days you are a SCL, attendance is required — if you miss such, you will get a zero as score; you will be able to replace that zero only if you are regularly attending class and have informed me, in advance, of your valid reason for missing that day.
In each grading category, the lowest n scores of that type will be dropped, where n is the value in the "# dropped" column. The total remaining points will be multiplied by a normalizing factor so as to make the maximum possible be 100. Then the different categories will be combined, each weighted by the "course %" from the following table, to compute your total course points out of 100. Your letter grade will then be computed in a manner not more strict than the traditional "90-100% is an A, 80-90% a B, etc." method. [Note that the math department does not give "+"s or "-"s.]
| # of such | # dropped | revision % | course % | |
|---|---|---|---|---|
| T&Qs: | ≈45 | 5 | 0% | 10% |
| CCT §§: | ≈4 | 0 | 75% | 15% |
| Homework: | ≈20 | 2 | 75% | 25% |
| Midterms: | 2 | 0 | 33.3% | 25% |
| Final Exam: | 1 | 0 | 0% | 25% |
Nota bene: Most rules on due dates, admissibility of make-up work, etc., will be interpreted with great flexibility for students who are otherwise in good standing (i.e., regular classroom attendance, homework (nearly) all turned in on time, no missing quizzes and tests, etc.) when they experience temporary emergency situations. Please speak to me — the earlier the better — in person should this be necessary for you.
Contact outside class: Over the years I have been teaching, I have noticed that the students who come to see me outside class are very often the ones who do well in my classes. Now correlation is not causation, but why not put yourself in the right statistical group and drop in sometime? I am always in my office, PM 248, during official office hours. If you want to talk to me privately and/or cannot make those times, please mention it to me in class or by e-mail, and we can find another time. Please feel free to contact me for help also by e-mail at jonathan.poritz@gmail.com, to which I will try to respond quite quickly (usually within the day, often much more quickly); be aware, however, that it is hard to do complex mathematics by e-mail, so if the issue you raise in an e-mail is too hard for me to answer in that form, it may well be better if we meet before the next class, or even talk on the telephone (in which case, include in your e-mail a number where I can reach you).
Contact inside class: Here are some useful hand gestures which can be used during class discussions (or lectures) for everyone to participate without the room becoming too cacophonous:
A request about e-mail: E-mail is a great way to keep in touch with me, but since I tell all my students that, I get a lot of e-mail. So to help me stay organized, please put your full name and the course name or number in the subject line of all messages to me. Also, if you are writing me for help on a particular problem, please do not assume I have my book, it is often not available to me when I am answering e-mail; therefore, you should give me enough information about the problem so that I can actually help you solve it (i.e., "How do you do problem number n on page p" is often not a question I will be able to answer).
Academic integrity: Mathematics is more effectively and easily learned — and more fun — when you work in groups. However, all work you turn in must be your own, and any form of cheating is grounds for an immediate F in the course for all involved parties. In particular, some assignments, such as take-home portions of tests, will have very specific instructions about the kinds of help you may seek or resources you may use, and violations of of these instructions will not be tolerated.
Students with disabilities: The University abides by the Americans with Disabilities Act and Section 504 of the Rehabilitation Act of 1973, which stipulate that no student shall be denied the benefits of education "solely by reason of a handicap." If you have a documented disability that may impact your work in this class for which you may require accommodations, please see the Disability Resource Coordinator as soon as possible to arrange accommodations. In order to receive this assistance, you must be registered with and provide documentation of your disability to the Disability Resource Office, which is located in the Psychology Building, Suite 232.
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| Jonathan Poritz (jonathan.poritz@gmail.com) | Page last modified: Saturday, 14-Jan-2012 19:25:38 MST |