Spring 2006

 Instructor: Joanna Ellis-Monaghan (please call me Jo)

Office: STE 217A

E-Mail: jellis-monaghan@smcvt.edu

Phone: 654-2660

Office Hours: By appointment MW 10:30-11:20.

Text: A Friendly Introduction to Number Theory, 3rd edition, by Joseph H. Silverman. There are on-line student resources for the text available at http://www.math.brown.edu/~jhs/frint.html.

Overview:  A big part of this class is learning how to do and then talk and write about mathematics.  If you do a beautiful piece of mathematics, you need to be able to explain to other people what you have done, and you need to be able to carefully write up your results so that other people can easily understand it.  Mathematics, and cutting edge research in almost any field, involves the steps: think of something cool, write stuff (e.g. small examples) down to make sure you are on track, discuss it with a few close friends/colleagues, write it up carefully, give a talk on it to a larger audience, write it up for publication.  Another extremely important component of doing mathematics is developing the ability to learn new mathematics independently from texts, journal articles, and colleagues.  This course is structured to give you some experience in the real thing.

Class notes: Since we have a smartboard equipped room, I will  post the class notes after each class here: Class Notes.  This is also where you will find careful write-ups of the exercises.

Class Web Page: You can find a link to the class web page on my home page at  http://academics.smcvt.edu/jellis-monaghan/ .  I will post class communications here (a copy of this handout for example). This also where I would post any practice exams or on-line resources. I will also use this page to notify you in the case of class cancellation, homework hints, answers to frequently asked project questions, or grade postings. You need to check it regularly.

Course Software: We will be using Maple, mostly for very simple things like testing if a number is prime, finding GCD's, factoring.  Pretty much every command you might need is discussed here: Number theory/maple_intro.mw.  

Collaborative groups: You will be working with a group of 5 collaborators in each class to address at least one exploratory exercise per chapter.  For each exercise, select

• two people to turn in the working notes,

• one to turn in the first draft,

• one to turn in the final draft (which will be scanned and posted on the website), 

• one to present your work to the rest of the class.

You are responsible for rotating these roles through your group so that by the end of the term each person has done each thing roughly the same number of times as every one else.  See the guidelines for collaborative work for what to turn in.  You are also encouraged to study/work together—prepare for exams, discuss homework and software applications, but each person must turn in their own work for the individual homeworks below.

Individual Homework: Homework is due two class periods after each chapter is finished.  For every individual homework question, you must suggest and try a follow up question to be doing better than average (B-/C+) work.  You will also chose one of these follow up questions as one of the substantial pieces of work for the course (see below)

Homework (both individual and collaborative)  must conform to the following:

1. Your name in the upper right hand corner of each page.
2. Stapled (not folded or paper-clipped) in the upper left hand corner.
3. Problems and sections in order, with answers checked.
4. Problems neatly done, no scratch-outs (doesn't apply to collaborative working notes), with all work shown! No credit for answers only.
5. Writing on one side of the paper only.
6. Most importantly, proofs must be carefully constructed, logically coherent, and grammatically correct.
7. The solutions must be presented professionally.  This means clear writing and explicit references to any theorems or definitions needed.  Good models for the style of writing required are the proofs and examples given in the text.

Self-study Guide:  Prepare Study Sheets for each chapter as it is covered. You will be able to use these on the Midterm and Final Exams, and they will also be collected/graded, so the more thorough, the better.  They should include all theorems, formulas, and definitions, with small examples attached to help you understand what is being said.  Remember that there is as much information in the exercises as in the chapter text.  There is no page limit here, but if it is longer than 3 or 4 pages, you might want to come up with a system to let yourself find things easily during a test.

 'Follow up' and Exploratory Questions: You must choose one follow on question, from either the collaborative or individual problems, and develop it fully (this does not necessarily mean that you solve it--see the guidelines).  You must also fully solve one of the exercises in any section we do not cover in class (i.e. any chapter except  1-12,28-32, 43-45).  No two people may work on the same problem (either follow up or Exploratory), so first-come-first-serve on the selections.  You must have selected your two problems by the beginning of class on Friday, April 7.  or (Note:  people who have taken abstract algebra must choose problems that do NOT overlap with the abstract algebra course, e.g. no RSA code, although you may use any tools you wish from abstract algebra or anywhere else for that matter to tackle the problems).  These may be turned in at any point in the course (the earlier the better if you would like feedback from me), but no later than the beginning of class on Monday, May 1.

Ron Graham Continuation Paper:  There are two required talks by eminent mathematician Ron Graham, one during class on Monday, March 6, and one, on the Mathematics of Juggling, in the Farrell room at noon on Tuesday, March 7. (You must let me know immediately if you have a conflict with this second talk).   The assignment is to write a paper, due Monday, March 27, about a piece of mathematics inspired by his life, work, talks, or conversations with him.  Guidelines.  There is a third, general audience talk, “Mathematics in the 21^st century: Problems and Prospects” in McCarthy Recital Hall at 7 PM on March 6.  You may attend this and write 2-3 paragraphs about it for some extra credit. Class will not meet the Wednesday, March 8, or Friday March 10, to allow time for these talks and research.

Midterm Exam:  There will be an inclass midterm exam on Friday, March 3.  You may use your study sheets and a calculator during the exam.

Final Exam: The exam will be on Monday, May 8, 9:00-11:30.   You may use your study sheets and a calculator during the exam.

Grading: Collaborative work--20%,   Individual homework--20%, Exploratory and follow up questions--20%,  Ron Graham continuation paper-- 10%,  Midterm (including study sheet)--12%, Final exam (including study sheet)--18%.

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