Number Theory for Poolesville Students

Fall 2013 -- Spring 2014, Dr. Ron Lipsman

The first class meeting will be on Tuesday, Sept 3, at 3:00 in Room 38. Starting the following week and continuing thereafter, the class will met on Thursdays at 3:00 in Room 38.

Text

To obtain a copy of the text, please see Ms. L. Loomis, Math Department, Poolesville High Schol. The Preface to the book, which you should consult, was inadverently omitted from the bound copies. Click here for a replica.

Prerequisites

The main prerequisite is a one-year course in one-variable calculus offered at Poolesville High School. Since most students will have taken the Magnet Analysis version, it will be assumed that the student is familiar with the contents of Chapters 1-10 in Calculus of a Single Variable (4th ed. or later) by Larson et al -- or the equivalent. In fact, the number of calculus results that we shall need is rather limited. The calculus requirement is there primarily to ensure that the student has the requisite sophistication to handle the topics in a college level number theory course. In particular, we will encounter far more in the way of proofs than is normally seen in a high school math course. In addition, it will be helpful if students will have been exposed to extra-calculus topics such as: mathematical induction, the binomial therem, the geometric series and Cantor's Theory of Sets. (In fact, the first three of these topics may be found in the Appendix of KW.) Finally, exceptionally well-qualified students who are currently enrolled in one-variable calculus at Poolesville may be admitted to the course with the approval of Ms. Loomis.

Exams

Content removed 3/26/19

Assignments

At the conclusion of each of the chapters in KW, there are three sets of problems: Homework Exercises, Projects and Computer Explorations. Each week approximately 5 Exercises will be assigned -- roughly 3 from odd-numbered problems and 2 from even-numbered problems. (You are free, and encouraged, to work additional problems from the text.) Hints for solutions and many answers to odd-numbered problems are found at the back of the text. You should consult those only after you have first attempted to solve the problem on your own. Solutions to the two even-numbered problems are to be submitted within THREE days, i.e., by 11PM on Sunday evening. You should write out your solution, scan it and email it (as a pdf or jpg) to: rlipsman@math.umd.edu. You may use no more than 1 sheet of paper per problem. Exercises that are to be submitted will be marked with an asterisk (*).

Over the course of the year, there will also be assigned 2 Projects and 3 Computer Explorations. The due dates are again by 11PM on Sunday evening, three days from the date of the assignment. These probems are much more substantial than the Exercises (so no paper limit). In addition, the use of MATHEMATICA (Mma) will be helpful for the former and essential for the latter. You are encouraged to submit a Mma Notebook (instead of a hand-written solution) for the Projects; and you are required to do so for the Explorations. See the Mathematica page for the policy on collaboration. See the Grading Policy below for point allocations.

Grading Policy

The maximum possible number of points you can score in this course is 650. Your final grade will depend upon the total points you score on the: KW Exercises; KW Projects; KW Computer Explorations (using MATHEMATICA), three hourly exams, and the final exam, distributed as follows:

CategoryTotal Points
Projects20
Computer Explorations30
Exercises100
Three Hourly Exams300
Final Exam200

There will be no make-up exams for any of the three in-class exams. If you miss ONE in-class exam (due to an excused absence), then your score for that exam will be one half of your final exam score. In all other cases if you miss one or more in-class exams, then no final grade will be issued for the course.

Finally, there will be several additional Projects and Explorations (likely five in total) listed as optional assignments. You may earn up to an additional 50 points for these; points that will count in the numerator, but not the denominator in computing your course score.

Class Schedule

The class meets on Thursdays, 3:00PM -- 4:00PM in room 38 at Poolesville High School.

Consultation Hours

Because of the unusual nature of the course, there is no opportunity to schedule office hours wherein students could ask questions or discuss course work outside of class with Dr. Lipsman. However, Dr. Lipsman welcomes phone calls, texts or emails from students at any time with relevant questions, comments or suggestions. Please feel free to avail yourself of this opportunity. Also, Dr. Lipsman tends to arrive early on Thursday and hang around a bit after class. If your class and transportation schedules permit, feel free to engage him at those times.

Nature of the Course

Since Number Theory is largely about the discovery of the remarkable properties of numbers (esp. integers, primes, rational numbers), its study typically involves proofs and the development of abstract methods. This differs from the study of Calculus or Ordinary Differential Equations (ODE) wherein the subject leads investigators towards symbolic and numeric algorithms for carrying out concrete processes. Thus this course will be much less formulaic than what you ecountered in calculus or -- if you participated last year -- differential equations. In the latter, the student learns many formulas and techniques that can serve as a foundation in more advanced courses in math and science. In this course, you will learn more about the kind of abstract reasoning that will also serve you well in those courses.

One consequence of this difference will be the role of mathematical software -- in our case, MATHEMATICA -- in the course. Mma is ideally suited to Calculus and ODE because it implements and automates the algorithmic processes that one encounters there. It is less ideally suited to proof construction and abstract reasoning. Nevertheless, Mma does have a rich set of number theory commands. Unfortunately, there is no analog of the Mma instruction book for differential equations that we used last year in ODE. Thus it will be largely left to the student to discover the relevant commands and how to use them. To get you started, go to the Help tab in a Mma Notebook, open the Documentation Center and search for "guide/NumberTheory".


Disclaimer: This page is updated once or twice per week during the school year. It is entirely possible that events have superseded the information contained herein. If so, announcements will be made in class. In urgent situations, e.g., snow cancellations, email will be sent. In any case, you may send me email inquiries.

Ron Lipsman
Professor Emeritus of Mathematics
Department of Mathematics
University of Maryland
College Park, MD 20742

Office: Mathematics Building, Room 4301
Phone: (301) 405-7061
Cell Phone: (301) 538-8381
Email: rlipsman@math.umd.edu
Backup Email: rlipsman@umd.edu