Math 350 (Number Theory) Fall 2015
Instructor:
Ching-Li Chai
Office: DRL 4N36, Ext. 8-8469.
Office Hours: M 10:00-10:50, and by appointments.
Email: chai@sas.upenn.edu
Grader:
The Gia Hoang
Email: theh@sas.upenn.edu
Office Hours: Th 3:30-4:30, and by appointments.
Homework Assignments
Suggested projects and the
current sign up list. You are welcome
and encouraged to find your own topic not on the suggested list.
The presentations are 12 minutes each, which we hope to begin in the middle of
November. The written report can be handed in at the time of the presentation,
but no later than the last day of classes in any case.
Presentation schedule
General Information
- Lectures: MWF 1--2 PM, DRL 4C6.
First meeting: Wednesday, August 26, 2015.
- Course description:
This is an introductory course to Number Theory, which is about
properties of whole numbers.
Abstract Algebra is NOT a required background, nor is calculus.
The concept of the big-O notation in calculs is explained in chapter
40 of the textbook by Silverman.
We will cover the traditional topics, including
- congruences,
- some polynomial congruence equations such as the Fermat equations,
- multiplicative functions,
- quadratic reciprocity and the Jacobi symbol,
- prime numbers and their distribution,
- applications of number theory to public key cryptography,
- continued fractions and Pell's equations.
Part of the goal of this course is to introduce the idea and practice
of rigorous mathematical proofs.
- Textbook: J. H. Silverman, A Friendly Introduction to
Number Theory, 4th edition, 2012.
A list of errors, together with some online-only chapters, are
available from
AFINT home page on Joe's site Although there are some
new exercises in the third edition, you can mostly get by with the
third edition. The differences between the third and the fourth
edition are described in the
AFINT home page.
This book is nice written and carfully paced, gentle at the
beginning to ease readers into the world of rigorous mathematics.
- There will be two in-class examsm, as well as a
report/presentation at the end of the semester.
- The COURSE GRADE is based on: Homework (40%), In-class Exams (40%),
Report/Presentation (20%).
- Some References:
- H. Davenport, The Higher Arithmetic, 1952. A classic.
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,
5th ed., Oxford University Press, Oxford, 1979.
A classic, and a wonderful introduction to analytic number theory.
- K. Kato, N. Kurokawa and T. Saito, Number Theory I, Fermat's
Dream, Amer. Math. Soc., 2000. A delightful introduction to
algebraic number theory at the graduate level.
- E. Landau, Foundations of Analysis, Chelsea, 1960.
Delightful treatment of the construction of integers, rational numbers
and real numbers, from the Peano axioms.
- E. Landau, Elementary Number Theory, Chelsea, 1958.
English translation of Landau's famous Elementare Zahlentheorie.
It gives a succinct treatment of number theory, including some
advanced topics such as the class number formula for quadratic
fields.
- I. Niven, H. Zuckerman and H. Montgomery, An Introduction to
The Theory of Numbers, 5th ed., 1991.
- Kenneth Rosen, Elementary Number Theory, Addison Wesley,
6th edition, 2010.
- Andre Weil, Number Theory for Beginners. A very short
presentation by a master.
- Andre Weil, Number Theory: An Approach Through History :
From Hammurapi to Legendre.
Authoritative history of number theory by a master.
- A few reference on crytography
- Johannes Buchmann, Introduction to cryptography,
Springer, 2001.
- Neal Koblitz, A Course in Number Theory and
Cryptography, graduate-level mathematical treatment.
- Douglas Stinson, Cryptography,
Theory and Practice, second edition,
Chapman and Hall/CRC, 2002.
- Bruce Schneier, Applied
Cryptography, second edition, John Wiley & Sons, 1996.
- Wade Trappe and Lawrence Washington,
Introduction to Cryptography
with Coding Theory, Prentice Hall, 2003.
Important Dates:
First day of class: Wednesday, August 26
Labor Day: Monday, September 7
Rosh Hashana, Monday September 14 and Tuesday September 15
Yom Kippur, Tuesday September 22 and Wednesday September 23
No class on Friday, September 25
Drop period ends: Friday, October 2
Fall break: October 8 (Thursday)-10 (Saturday)
In-class exam: Monday, October 26, in class
Last day to withdraw: Friday, November 6
Thanksgiving: November 24- November 25
Last day of class: Tuesday December 8
Reading days: December 9 (Wednesday)-December 10 (Thursday)
In-class exam: Monday December 7 (in class).
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