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

Instructor -- Pavel Guerzhoy

The class meets Tuesdays and Thursdays, 12 -- 1:15 pm at 414, Keller Hall.

Office: 501, Keller Hall (5-th floor)

e-mail: pavel(at)math(dot)hawaii(dot)edu (usually, I respond to e-mail messages within a day)
Office hours: Tuesdays and Thursdays 2-3pm
Ofiice hours will be conducted online till the end of the semester.
Please e-mail me your request, and you will get back an invitation to Google Meet .

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 due Thursday, May 14 at 2pm 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.

    Date Sections and Links to online video lectures Homework Assignment and additional texts and lecture notes Writing Homework Assignement
    Tue, 14 Jan Ch. 1-1 1-12,17,18 on p6
    Thu, 16 Jan Ch. 1-2 1-7 on p10
    Tue, 21 Jan Ch. 2-1,2-2 1,2,3 p14
    Thu, 23 Jan Ch. 2-2,2-3,2-4 1,2 on p21; 1,2,3,6 on p25; 1-12 on p28 additional text HW1 due Friday,1/24
    Tue, 28 Jan Ch. 4-1,4-2 1-5,7 on p51; 1-4 on p55
    Thu, 30 Jan problem session
    Tue, Feb 4 Ch. 5-1,5-3 1-3 on p61; 1-6 on p70
    Thu, Feb 6 Ch.5-2 1-23 on p63
    Tue Feb 11 problem session /review HW1 redo
    Thu Feb 13 quiz
    Tue Feb 18 Ch. 3-1 1-10 on p33
    Thu Feb 20 Ch. 3-4 1-4 on p43 HW2 due
    Tue Feb 25 generating functions 5-8 on p43
    Thu Feb 27 generating functions additional text
    Tue Mar 3 generating functions HW2 redo
    Thu Mar 5 Bernoulli numbers
    Tue Mar 10 Riemann's zeta-function additional text
    Thu Mar 12 Ch.6-1,6-2,6-3 1,4,6,8-11,13,14,15 on p81, 1-5,8-12 on p84
    From this point on the class moves online!
    Tue Mar 24 Ch.6-4
    part 1
    part 2
    part 3
    notes of the online lecture (Mar 24 and Mar 31)
    full lecture on youtube with subtitres available
    HW3 due Friday Mar 27
    the due date has been extended by students' requests
    Tue Mar 31 Ch.6.4 1-4,7,8,11 on p90-91
    Thu Apr 2 Ch 6.4
    notes of the online lecture
    full lecture on youtube with subtitres available
    Tue Apr 7 problem session notes to problem session on Section 6-1
    problem 1
    problem 4
    problem 11
    problem 6
    problems 8,9,10
    Thu Apr 9 problem session notes to problem session on Section 6-2
    problem 1
    problems 2,5,9
    notes to problem session on Section 6-4
    problems 1,3
    problem 11
    Tue Apr 14 additional text
    notes of the online lecture
    full lecture on youtube with subtitres available
    HW3 redo due Tuesday Apr 14
    Thu Apr 16 Ch.7-1,7-2 1-7 on p96; and 4,7,9,11,14,15 on p98
    notes of the full lecture
    lecture on 7-1,7-2, I
    lecture on 7-1,7-2, II
    lecture on 7-1,7-2, III
    Tue Apr 21 quiz
    Thu Apr 23 problem session notes to problem session on Section 7-1
    Problem Session on Section 7-1

    notes to an EXTENSION to problem session on Section 7-1
    Problem Session on Section 7-1 extension
    HW4 due Friday April 24
    Tue Apr 28 problem session notes to problem session on Section 7-2
    Problem Session on Section 7-2
    quiz due
    Thu Apr 30 continued fractions I additional text
    notes to all videos on continued fractions
    Tue May 5 continued fractions II lecture on continued fractions I
    lecture on continued fractions II
    lecture on continued fractions III
    lecture on continued fractions IV
    HW4 redo due
    Thu May 7 continued fractions III APPLICATION of continued fractions
    HW4 redo
    DUE THURSDAY, MAY 14 at 2pm