General Information
and Course Policies

Course Overview. Patterns within the set of natural numbers have enticed mathematicians for well over two millennia, making Number Theory one of the oldest branches of mathematics. Still, numerous number theoretic problems remain open to this day, and many of these problems continue to entice the mathematical masses. In this course we will explore some of the classical problems in number theory, focusing largely on open problems and partial results related to these problems. Our approach will be somewhat different from most upper level math courses in that the second half of the semester will be devoted to the reading and presenting of research papers. (That is, students will be responsible for reading and presenting these papers to one another.) We will need to understand the basic notions of number theory before tackling this feat, however, so we will spend the first half of the semester working through Chapters 1-18 of Joseph Silverman's textbook A Friendly Introduction to Number Theory. In doing so, we will cover many of the topics typically included in a standard number theory course: divisibility, primes and their distribution, congruences, arithmetic functions, RSA cryptography and more.

The Textbook. Joseph H. Silverman, A Friendly Introduction to Number Theory 3rd ed., Pearson Prentice Hall. You might be able to find the book at a cheaper price online by visiting amazon.com:
http://www.amazon.com/gp/product/offer-listing/0131861379/ref=buy_pb_n_//103-5773887-7491824?condition=new

Grades. Your grade will be based on your performance on daily homework, your paper presentations (which includes a written paper), a midterm, and a final exam. Each will be weighted as follows.

% of Total

Daily Homework and Presentations/Participation

25

Paper Presentations

30

Midterm

25

Final Exam (take-home)

20

TOTAL:

100

Daily Homework. Throughout the first half of the semester, you will be given daily homework assignments that will consist largely of problem-solving and proof-writing. Sometimes I will collect and grade this homework, and other times students will present their work in class. Some of the exercises will come directly from the text and others will be written by your professor.

Your homework during the latter half of the semester will be significantly different in that you will be spending your time reading and understanding the research papers presented by your classmates. NOTE: You will be expected to read each research paper prior to the day it is to be presented.

HOMEWORK POLICY

  1. Homework is due at the START of class on the assigned due date, unless your professor specifies otherwise. Late homework will not be accepted. If you know you will be missing class for some reason (e.g., an athletic event), turn in your assignment BEFORE you leave. Under extenuating circumstances extensions may be granted, but this should be discussed with your professor in advance.
  2. Your homework will be evaluated on neatness, completeness, and correctness.
  3. Group work is encouraged, but assignments must be written up INDIVIDUALLY unless you are told otherwise. Copied work will receive no credit - even if the work was discussed in collaboration with a classmate before write-up. See "Academic Honesty" below for more information.

Paper Presentations. During weeks 10-14, students will read and present papers to one another. The reading list will be determined by both the students (to ensure that there is ample student interest in the topics covered) and the instructor (to ensure that the papers chosen are accessible to the students).

The Paper Presentation will have three components:

1. Paper Proposal and Outline - Due Tuesday, November 1
2. Written Paper - Due Tuesday, November 14
3. Paper Presentation - To be decided (sometime between Nov. 14 and Dec. 13)

Paper Proposal and Outline:
Each student will be responsible for finding a paper and discussing its appropriateness with the professor. Given the nature of mathematical research, it is important that you make sure in advance that your paper is accessible. After this initial approval, you will write a proposal that describes the purpose and direction of the paper you will write (which, in turn, will provide the basis of your paper presentation to follow.) Your proposal should be 2-3 pages long, including an introductory discussion, an outline of the theorems and proofs to be covered, and an indication of the importance/relevance of the theory involved. It should also include a bibliography.

Written Paper:
The paper component of this assignment will serve as a written version of the presentation that you will ultimately give to the class. However, your written paper will be more detailed than your actual presentation, as you will not have enough time during your presentation to cover everything in full. For example, in your paper you will want to include all details of any proof you will discuss, and you will want to include a significant level of discussion to support and motivate the theorems and proofs along the way. The length of the papers will vary from one student to the next, depending on the nature of the material to be covered and the amount of detail or graphics included. However, I'm guessing that they will be approximately 5 pages long for students working alone and 10 pages long for students working in pairs.

Paper Presentation:
Each student will have 1/2 of a lesson between Nov. 14 and Dec. 13 to present his or her paper. This could be accomplished in one of two ways. In the first way, a student works alone and has 35-40 minutes to cover an entire paper (or a cohesive portion of a paper). Of course, in this situation the student would probably be working with a shorter paper. In the second case, two students work together to cover a single paper. In this case, the two students work together on all three components of the paper presentation and would have an entire lesson between the two of them to present their paper. This latter option is preferable because it allows the class to focus on one main idea or set of ideas per lesson, and it allows the presenters more freedom to cover the material at an adequate level.

Please feel free to get advice from your professor at every step along the way on this assignment. Most of you will be reading and presenting mathematical research papers for the first time. You will likely have a lot of questions. Be assured that you will learn a great deal during this experience, and what you learn will prove helpful to your future work (e.g., students from my last offering of this class tell me they found this assignment to be very helpful to them when it came time to complete their senior exercise in mathematics.)
The Midterm. The mid-term exam will have two components: an in-class component and a take-home component. The in-class component will be held on Thursday, October 6. The take-home component will be distributed on the same day and will be due at the beginning of class on Thursday, October 13. You will choose a 48-hour period between the end of class on October 6 and the beginning of class on October 13 to complete the exam. (Note: October 8-9 are Reading Days. Plan accordingly.) Both components of the midterm will cover the material discussed during the first six weeks of the semester (see the syllabus for Block I).
The Final Exam. The final exam will be take-home and cumulative. It will be distributed on December 13 (the last day of class) and will be due on December 19 at 8:30am (as dictated by the College's Final Exam Schedule.) You will choose a consecutive 72-hour period during this time frame to complete the exam.
Academic Honesty. In general, the rules set forth in the 2005-2006 Course of Study apply. Presenting the work of others as your own is strictly prohibited. In the case of homework, you may collaborate with others in discussing how a problem may be solved, but your write-up must be your own. If you submit work that contains the ideas or words of someone else, then you must provide proper citation. Assistance can not be given nor received (other than by the instructor) on any quiz, or exam associated with this course, except where explicitly allowed by the instructor. In the case of a group assignment, all members of the group should contribute equally to writing the final product. And every member of the group is responsible for the content of the entire paper, not just the section(s) that are written by that person. Don't put your name on a paper written by others. For further information, consult your instructor.
Learning Disabilities. If you have a disability which requires an accommodation in this class, please feel free to discuss your concerns with me, but you should also consult Ms. Erin Salva, (Coordinator of Disability Services; Office of the Dean for Academic Advising, PBX 5453) as soon as possible. Ms. Salva (in consultation with the L.E.A.R.N. committee) has the authority and the expertise to decide on the accommodations that are proper for your disability. Though I am happy to help you in any way I can, I cannot make any accommodations for learning (or other) disabilities without proper authorization from Ms. Salva.
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