Math 420(W), Introduction to the Theory of Numbers

Instructor -- Pavel Guerzhoy

The class meets Tuesdays and Thursdays, 10:30 -- 11:45 pm at 413, Keller Hall.

Office: 501, Keller Hall (5-th floor)
tel: (808)-956-6533
e-mail: pavel(at)math(dot)hawaii(dot)edu (usually, I respond to e-mail messages within a day)
Office hours: Tuesdays and Thursdays 2:30-4:30pm

General Expectations
The Department of Mathematics has a general expectations statement, which we are assumed to follow in this class.
In this class we use the book
  • George E. Andrews, Number Theory, Dover Publications, Inc., New York
    The book is really unavoidable, and cannot be replaced with another textbook!

    There are, however, many number theory textbooks which may be useful. Just several samples are provided below.

  • William J. LeVeque, Elementary Theory of Numbers, Dover Publications, Inc., New York, is of particular interest. It contains several additional topics along with alternative approaches to the some theorems under the consideration.
    The following books are much harder, of more advanced level, and cover much bigger amounts of material. They may be recommended for further reading.
  • Serre, J.-P., A course in arithmetic, Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg. This book is far from being elementary. It is written by the best mathematician of the last century, and is an extremely valuable reading for those who want to specialize in number theory.
  • Ireland, Kenneth; Rosen, Michael, A classical introduction to modern number theory, Second edition. Graduate Texts in Mathematics, 84. Springer-Verlag, New York, 1990. This is an intermediate level book, which covers various important topics, and may serve as a bridge between elementary and advanced number theory.
    Course Objective
    To learn some ideas and notions of elementary number theory. In some cases, we concentrate on the rigorous mathematical proofs; in other cases, we concentrate on properties of the objects, and ideas involved into their investigation. The class is designated as writing-intensive. As a consequence, mathematical writing, particularly, the writing of clear and correct proofs is a subject of emphasis.
    Intro to Advanced Mathematics, Math 321 or consent.

    Grading Policy
    The course contains a combination of concepts, ideas and techniques, which the students must be able to apply in solving specific problems. A majority of these problems requires to either prove or disprove a certain statement. Since this is a writing-intensive class, we simultaneously learn how to write mathematical texts.

    At the end of the day, your grade will reflect your ability to solve specific problems. This assumes your ability to read and understand the textbook and supplemantary texts. To understand, in this context means to be able to create arguments which are similar to those provided in the texts. To create an argument, in this context, means to be able to write it down properly. More specifically, the following rules are to be taken.

  • Final exam will count for 30% of the final grade. The exam is cumulative (it covers all the material).
  • Four(?) writing homework assignments count together for 40% of the final grade.

  • Two midterm quizzes. The average grade for the quizzes counts for 30% of the final grade.

    Contents and regular Homework assignments

    This table is approximate, and will be updated regularly.

    Nov 24Nov 26
    Date Sections Homework Assignment Writing Homework Assignement
    Tue, 25 Aug Ch. 1-1 1-12,17,18 on p6
    Thu, 27 Aug Ch. 1-2 1-7 on p10
    Tue 1 Sep Ch. 2-1,2-2 1,2,3 p14
    Thu 3 Sep Ch. 2-2,2-3 1,2 on p21; 1,2,3,6 on p25 additional text
    Tue 8 Sep Ch. 2-4 1-12 on p28
    Thu 10 Sep problem session
    Tue 15 Sep Ch. 4-1,4-2 1-5,7 on p51; 1-4 on p55 HW1
    Thu 17 Sep Ch. 5-1,5-2 1-3 on p61; 1-23 on p63
    Tue 22 Sep Ch.5-3 1-6 on p70 HW1 redo
    Sep 24 problem session /review
    Sep 29 problem session /review
    Oct 1 quiz
    Oct 6 Ch. 3-1 1-10 on p33
    Oct 8 Ch. 3-4 1-4 on p43
    Oct 13 Ch. 3-4 5-8 on p43
    Oct 15 generating functions additional text
    Oct 20 generating functions
    Oct 22 Bernoulli numbers HW2
    Oct 27 Riemann's zeta-function additional text
    Oct 29 Ch.6-1,6-2,6-3 1,4,6,8-11,13,14,15 on p81, 1-5,8-12 on p84
    Nov 3 Ch.6-4 1-4,7,8,11 on p90-91 HW3
    Nov 5 problem session additional text
    Nov 10 problem session HW2 redo
    Nov 12 Ch.7-1,7-2 1-7 on p96; and 4,7,9,11,13,14,15,16 on p98
    Nov 17 problem session
    Nov 19 problem session HW3 redo
    problem session
    quiz HW4
    continued fractions I additional text
    continued fractions II
    continued fractions III
    problem session
    review for the final exam I
    review for the final exam II