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

Instructor -- Pavel Guerzhoy

The class meets Tuesdays and Thursdays, 10:30 - 11:45am at 403, 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 12:00-1:20 and 3:00-4:00pm

Final Exam

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.
    A first course in linear algebra, Math 311 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. To understand, in this context means to be able to create arguments which are similar to those provided in the text. 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.
  • Three 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.

    An example of how to read the table.

    Date Sections Homework Assignment Writing Homework Assignement
    Tue, 12 Jan Ch. 1-1 1-12,17,18 on p6
    Thu, 14 Jan Ch. 1-2 1-7 on p10
    Tue, 19 Jan Ch. 2-1,2-2 1,2,3 p14
    Thu, 21 Jan Ch. 2-2,2-3 1,2 on p21; 1,2,3,6 on p25 8 on p7; 4 on p11
    Tue, 26 Jan Ch. 2-4 1-12 on p28
    Thu, 28 Jan problem session
    Tue, 2 Feb quiz
    Thu, 4 Feb Ch. 4-1,4-2 1-7 on p51; 1-4 on p55
    Tue, 9 Feb Ch. 5-1,5-2 1-3 on p61; 1-23 on p63
    Thu, 11 Feb Ch.5-3 1-6 on p70
    Tue, 16 Feb problem session /review
    Thu, 18 Feb Ch. 3-1 1-10 on p33
    Tue, 23 Feb Ch. 3-4 1-4 on p43
    Thu, 25 Feb Ch. 3-4 5-8 on p43
    Tue, 2 Mar generating functions additional text 9,15,11,12 on p64
    Thu, 4 Mar Bernoulli numbers
    Tue, 9 Mar Riemann's zeta-function additional text
    Thu, 11 Mar quiz quiz
    Tue, 16 Mar generating functions
    Thu, 18 Mar Ch.6-1,6-2,6-3 1,4,6,8-11,13,14,15 on p81, 1-5,8-12 on p84
    Tue, 30 Mar Ch.6-4 1-4,7,8,11 on p90-91
    Thu, 1 Apr problem session additional text
    Tue, 6 Apr Ch.7-1,7-2 1-7 on p96; and 4,7,9,11,13,14,15,16 on p98 1 on p86
    Thu 8 Apr problem session
    Tue, 13 Apr quiz
    Thu, 15 Apr continued fractions I additional text 13,14 on p99
    Tue, 20 Apr continued fractions II
    Thu, 22 Apr continued fractions III
    Thu, 27 Apr problem session
    Thu, 29 Apr review for the final exam I
    Thu, 4 May review for the final exam II