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

Instructor -- Pavel Guerzhoy

The class meets Tuesdays and Thursdays, 3 - 4:15pm at 402, 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 12:00-2:45pm

AMS Spring Western Section Meeting March 3-4, 2012

Hawai‘i Conference in Algebraic Number Theory, Arithmetic Geometry and Modular Forms, March 6-8, 2012

For both conferences, I recommend put special ettention at:

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 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.
  • 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.

    Date Sections Homework Assignment Writing Homework Assignement
    Tue, 10 Jan Ch. 1-1 1-12,17,18 on p6
    Ch. 1-2 1-7 on p10
    Ch. 2-1,2-2 1,2,3 p14
    Ch. 2-2,2-3 1,2 on p21; 1,2,3,6 on p25
    Tue, 24 Jan Ch. 2-4 1-12 on p28 8 on p7; 4 on p11
    problem session
    Thu, 2 Feb quiz
    Tue, 7 Feb Ch. 4-1,4-2 1-5,7 on p51; 1-4 on p55 resubmission of the first homework
    Ch. 5-1,5-2 1-3 on p61; 1-23 on p63
    Ch.5-3 1-6 on p70
    problem session /review
    Ch. 3-1 1-10 on p33
    Ch. 3-4 1-4 on p43
    Tue, Feb 27 Ch. 3-4 5-8 on p43
    Thu, Mar 1 generating functions additional text
    Tue, Mar 6 Colloquium at 3:30 in Keller 401 All invited to attend! Matilde Marcolli
    Thu, Mar 8 Bernoulli numbers 9,15,11,12 on p64
    Tue, Mar 13 Riemann's zeta-function additional text
    quiz quiz Due date: Apr 3
    Ch.6-1,6-2,6-3 1,4,6,8-11,13,14,15 on p81, 1-5,8-12 on p84
    Ch.6-4 1-4,7,8,11 on p90-91
    problem session additional text
    Ch.7-1,7-2 1-7 on p96; and 4,7,9,11,13,14,15,16 on p98
    problem session
    problem session
    Thu, Apr 12 problem session 1 on p86
    Tue, Apr 17 quiz quiz
    Thu, Apr 19 continued fractions I additional text 13,14 on p99
    continued fractions II
    continued fractions III
    problem session
    review for the final exam I
    review for the final exam II