Math 350 (Number Theory) Spring 2017

Instructor: Ching-Li Chai

Office: DRL 4N36, Ext. 8-8469.
Office Hours: M 12:00-1:30, and by appointments.
Email: chai@sas.upenn.edu

Grader: Umair Mian Ahsan
Email: ahsanm@math.upenn.edu
Office Hours: Tuesday 5:00-6:30 pm, DRL 3N2D.

Homework Assignments

Suggested projects and the current sign up list.
presentation schedule
You are expected to "sign up" for a topic before the end of Janury, so that you will have ample time to prepare. You are encouraged to find your own topic not on the suggested list. The presentations are about 10 to 12 minutes each, which will begin in the second week of April. 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.

General Information

• Lectures: MF 1:30--2:50 PM, DRL 4C4.
  First meeting: Wednesday, January 11, 2017 (which is a University Monday)
  
  
• 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 fourth edition, you will do fine with third edition. (You can get a used copy of the 3rd edition at amazon for under $20. A new copy of the 4th edition costs $147.) 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 exams, 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. Roughly at the same level of this course, more concise than Silverman's text. Contains a treatment of quadratic forms and class numbers. After reading this book you have learned the mathematical contents in Gauss's famous Disquisitiones Arithmeticae.
  • 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. The material is in between this course and a graduate course in number theory. A chapter on elliptic curves was added in the 6th edition.
  • H. Hasse, Number Theory, 3rd ed., 1969. A classic. The first chapter, pages 1-103, covers the majority of the mathematical content treated in this course. The exposition is discursive and very readable, quite different from the Satze-Bewies format found in Landau. You need some abstract algebra (at the level of math 370) to fully profit from this book.
  • L. K. Hua, Introduction to Number Theory, Springer-Verlag, 1982. A classic which influenced generations of Chinese mathematicians. Like Hardy-Wright, this book can be profitably read by anyone with a good grasp of calculus. Covers more ground than Hardy-Wright (5th edition).
  • 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 at the level of Davenport's book, plus some more advanced topics such as the class number formula formula for quadratic forms and Dirichlet's L-functions for quadratic characters.
  • I. Niven, H. Zuckerman and H. Montgomery, An Introduction to The Theory of Numbers, 5th ed., 1991. The materials covered is comparable to Silverman's text but goes somewhat faster.
  • H. Rademacher, Lectures on Elementary Number Theory, Blaisdell, 1964. This thin (146 pp.) book is a jem, which ontaining a lot more than what is covered in this course, including Gauss's construction of the regular heptadecagon, proof of the quadratic reciprocity law for the Legendre symbol and Jacobi symbol via Gauss sums, Dirichlet's theorem on prime in arithmetic progressions, and a theorem of Brun on twin primes by the sieve method.
  • Kenneth Rosen, Elementary Number Theory, Addison Wesley, 6th edition, 2010. The level is similar to Silverman's text.
  • J.-P. Serre, A course in arithmetic. A concise masterful presentation of the arithmetic of rational numbers, including p-adic fields and Hasse principle for quadratic forms over rational numbers, Dirichlet's theorem on primes in arithmetic progressions and modular forms for the full modular group, all in about 110 pages.
  • Andre Weil, Number Theory for Beginners. A very short presentation by a master. Covers all basic materials (other than those related to computation and cryptography) in 70 short pages.
  • Andre Weil, Number Theory: An Approach Through History : From Hammurapi to Legendre. Authoritative history of number theory by a master.
  • A few references 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, January 11
Martin Luther King Jr. Day, Monday January 16, no class
Drop period ends: Friday, February 17
Spring break: March 5 (Sunday)-11 (Saturday)
In-class exam: Monday February 27, in class
Last day to withdraw: Friday, March 24
In-class exam: Monday April 24
Last day of class: Monday April 24
Reading days: April 27 (Thursday)-April 28 (Friday)

Some Links

• The Number Theory Web
  
  
• The primes page
  
  
• National Institute of Standards and Technology, Computer Security Resource Center
  
  
• Frequently Asked Questions About Today's Cryptography, from RSA Lab
  
  
• cryptography pointers