# Math 350 (Number Theory) Spring 2022

Instructor: Ching-Li Chai
Office: DRL 4N36, Ext. 8-8469.
Office Hours: M 11:15-12:00, and by appointments.
Email: chai@math.upenn.edu

Office:
Office Hours:

Homework Assignments

A list of possible projects. You are welcome and encouraged to find your own topic not on this list. But you are not supposed to do a project on a topic you already know well or have taken a course on. (We use an honor system here.)

#### General Information

• Lectures: MWF 10:15-11:15 am, 4C6 DRL
First meeting: Wednesday, January 12, online by zoom.
meeting ID: 986 8602 4473
passcode: 154042
URL: https://upenn.zoom.us/j/98686024473?pwd=MzcyeVZ6Qk11Zm1IV2RTdzVhR0JYdz09

• Textbook: Martin Weissman, An Illustrated Theory of Numbers, American Mathematical Society, 2017.
The exposition in this book stresses "visualization", and is quite different from all other textbooks I know of. Many students find the explanation easy to understand. It is also a nice coffee table book (for nerdy guests). The core materials of math 350 are contained in parts I and II of Weissman. They include Euclidean algorithm, prime factorization, Gaussian numbers, congruences, primitive roots modulo a prime number, linear and quadratic congruence equations, quadratic reciprocity, Legendre and Jacobi symbols.

• 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. We will cover the traditional topics, including
• the Euclidean algorithm
• prime factorization
• congruences and the modular arithmetic
• some polynomial congruence equations such as the Fermat equations,
• quadratic reciprocity and the Jacobi symbol,
Part of the goal of this course is to introduce the idea and practice of rigorous mathematical proofs.

• Homework will be assigned every week. You will present your solutions of some of the homework problems in class.

• There will be two in-class exams, plus a report/presentation at the end of the semester, on a topic you choose.

• The COURSE GRADE is based on: homework (35%), presentation of homework problems (10%), in-class exams (35%), report/presentation of your project (20%).

Important Dates:

• First day of classes: Wednesday, January 12
• MLK Jr. Day: Monday, January 17
• In-class exam: Friday, February 18, in class
• Drop period ends: Monday, February 21, no class
• Spring break: March 5 (Saturday)-13 (Sunday)
• In-class exam: Friday, March 25
• Last day to withdraw: Monday, March 28
• Last day of classes: Wednesday, April 27
• Reading days: April 28 (Thursday)-May 1 (Sunday)

Some References:

• G. Andrews, Number Theory, revised edition, Dover, 1994.
• A. Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, 1984. This small and thin (95 + xiii pages) textbook covers all standard topics in elementary number theory, including quadratic reciprocity, integral quadratic forms, Pell equation, continued fractions, plus short introductions to diophantine approximation, quadratic fields and diophantine equations. There is also an introduction to Gauss's Disquitiones Arithmeticae.
• H. Davenport, The Higher Arithmetic, 1952. This classic provides an extraordinarily lucid account of elementary number theory (including congruences, quadratic reciprocity and quadratic forms) in 170 pages. Can be read by anyone knowing high school algebra.
• 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.
• H. Hasse, Number Theory, 3rd ed., 1969. 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.
• L.K. Hua, Number Theory, Springer, 1982. Written in a style close to Landau, Hardy/Wright and Davenport, and covers more ground than Hardy/Wright, Davenport's Higher Arithmetic and Landau's Elementary 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, the first 4/5 of volume 1 of Landau's 3-volume Vorlesungen uber Zahlentheorie. It gives a succinct treatment of number theory, including some advanced topics such as the class number formula for quadratic fields. (Don't be fooled by the adjective "elementary" in the title. It is written in the "Landau" style, a relentless march of theorem, proof, theorem, proof, theorem, proof, with all theorems numbered consecutively. The last theorem of volume 3 is Satz 1046.)
• B. Mazur and W. Stein, Prime Numbers and the Riemann Hypothesis , CUP, 2016.
• I. Niven, H. Zuckerman and H. Montgomery, An Introduction to The Theory of Numbers, 5th ed., 1991.
• P. Ribenboim, The Little Book of Bigger Primes, 2nd ed., Springer, 2004.
• Kenneth Rosen, Elementary Number Theory, Addison Wesley, 6th edition, 2010.
• M. Schroeder, Number Theory in Science and Communication, 5th ed., Springer, 2009.
• J.-P. Serre, A Course in Arithmetic, Springer, 1973. An introductory graduate-level textbook by a master. The expositions are optimized: both succinct and crystal clear.
• I. M. Vinogradov, Elements of Number Theory, Dover reprint of an English translation of the 5th Russian edition. Written a famous analytic number theorist, it contains an exposition of all core materials of this course, plus plenty of exercises in 132 pages. The exercises are the best part of this book. There are also 84 pages of solutions for the majority of exercise problems.
• Andre Weil, Number Theory for Beginners. A very short (68 pages) presentation of the core materials by a master. The concepts of equivalence relations, commutative groups, cosets, rings, fields and polynomials are introduced in Weil's book, to put elementary number theory in the framework of abstract algebra. (It will be hard to imagine Weil not doing so.) Has plenty of exercise problems (but no solutions).
• 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.
Comments: Davenport's Higher Arithmetic, Hardy/Wright, Landau's Elementary Number Theory, and Hua's Number Theory are written in the classical style, and accessible to anyone with some exposure to calculus. (Calculus is not needed for Davenport's book, the shortest among the four. Complex analysis is used in a few sections (not chapters) of Hua's book, the longest among the four.) They are all great; which one you like better depends on your taste. If you have taken abstract algebra and analysis, Serre's A Course in Arithmetic and Kato/Kurosawa/Saito will lead you quickly to a glimpse of modern number theory. If you want a concise textbook to supplement Weissman, Baker or Vinogradov may be what you are looking for. And if you want a supplement which treats elementary number theoretic through basic ideas in abstract algebra, try Weil's small book.