MAT 311
Number Theory

Instructor:

Sorin Popescu (office: Math 3-109, tel. 632-8255, e-mail sorin at math.sunysb.edu)

Grader:

Caner Koca (office: Math 3-118, e-mail caner at math.sunysb.edu)

Schedule:

TuTh 02:20pm-03:40pm, Lgt Engr Lab 154

Prerequisites:

Either MAT 312 (Applied algebra), or MAT 313 (Abstract Algebra) or MAT 318 (Classical Algebra) are mandatory prerequisites for this class. In general some basic algebra exposure is required and assumed, but I will try to keep prerequisites to a minimum.

Textbook(s):

We will be covering a number of elementary topics in number theory and some applications. Some theoretical aspects will be trated in detail while others will provide brief but insightful and motivating excursions into topics like Mersenne Primes, number sieves, RSA cryptography, elliptic curves, etc if time will permit.

There are many excellent undergraduate books on the subject. Here is a sample list (all of them available in our library), however we will mainly use this semester only the first two of them:

The higher arithmetic, by H. Davenport (7th edition) Elementary Number Theory and Its Applications , by K. Rosen, (5th edition) An Introduction to the Number Theory, H.M. Stark Number Theory, G.E. Andrews Introduction to Analytic Number Theory, T.M. Apostol An Introduction to the Theory of Numbers, I. Niven and H.S. Zuckerman A Classical Introduction to Modern Number Theory, K. Ireland and M. Rosen Fundamentals of Number Theory, W.J. LeVeque Number theory with computer applications, R. Kumanduri and C. Romero

These are a mixture of classical texts (for example Dirichlet), modern efforts, more elementary (for example Kumanduri and Romero) and more advanced (for example Rosen, or Ireland and Rosen), algebraic (for example Andrews or Davernport) or analytic approaches (for example, Apostol). This course will concentrate only on elementary algebraic number theory, and applications.

Course description:

We will cover only parts of the textbook(s) (Davenport and Rosen) and the schedule may/will be adjusted based on students' preparation and progress.

Topic	Sections in textbook	Week	Notes
Numbers, sequences, and sums. Induction. Fibonacci numbers. Divisibility	Davenport Chap. 1	1/23-1/28
Greatest Common Divisor, Euclidean algorithm, Fundamental theorem of arithmetic		1/30-2/4
Linear Diophantine equations / Congruences		2/6-2/11
Fermat's little theorem / Euler's Formula	Davenport Chap. 2	2/13-2/18

Projects, Homework & Grading:

Homework and project (TBA) are an integral part of the course. Problems marked with an asterisk (*) are for extra credit. In addition to homework you will be required to hand in a research/scholarship/computing project. Projects with a nontrivial writing component may be used to satisfy the Mathematics Upper Division Writing Requirement.

Your grade will be based on the weekly homeworks (20%), project (25%), midterm (25%), and the final exam (30%). The two lowest homework grades will be dropped before calculating the average.

• Homework assignments
• Projects (updated)

The midterm will be held in class on 03/23.

A review session will be held on 04/28 in Math P-131, 3:00-4:00pm. The final exam will be comprehensive.

Grades are now posted on the Solar system. Have a nice summer!

Links:

The following is a short list of web sites devoted to number theory or number theoretic related topics relevant for our class:

• An On-Line Encyclopedia of Integer Sequences.
• Fibonacci Numbers and Nature. Or Tony Phillips' "The most irrational number". Also "Who was Fibonacci?": a brief biography of Fibonacci.
• Primes: Lots of interesting facts about prime numbers.
• Mersenne Primes: interesting facts about Mersenne primes, perfect numbers, and related topics.
• Primes is P: about a recent polynomial time deterministic algorithm to test if an input number is prime or not.
• RSA: The RSA company's web page containing lots of interesting information about the RSA public key cryptosystem and cryptography in general, from both a technological and a socio-political viewpoint.
• The RSA factoring challenge.
• The Wikipedia entry for RSA cryptography.
• The Enigma machine.
• Handbook of Applied Cryptography, by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone. It is a handbook for both novices and experts, introducing practical aspects of both conventional and public-key cryptography.
• The GCHQ Challenge (as of July 2005).
• Ron Rivest's Cryptography and Security links page.
• MIT Lecture Notes on Cryptography, by S. Goldwasser and M. Bellare.
• A brief history/introduction to error-correcting codes Digital Revolution - Error Correction Codes, Joseph Malkevitch, in What's New in Mathematics Feature Column of the AMS.
• Anatomy of Credit Card Numbers: a discussion of how to determine if a given credit card number might be valid or not.
• The Wikipedia entry for ISBN (that is International Standard Book Numbers). However, starting January 1, 2007, the book industry will begin using 13 digit ISBNs to identify all books in supply chain.
• A summary describing how information is encoded on Compact Discs. CDs use a modified form of the Reed-Solomon code called the Cross Interleave Reed-Solomon Coding (CIRC). Here is a short description of how Reed-Solomon codes are used for error-correction on a audio CD (Red Book Standard).
• FactorWorld is a web site dedicated to integer factorization results and algorithms. It also includes a list of recent factoring records.
• The list of the 221 currently known proofs of the Quadratic Reciprocity Law (from Legendre (1788) and Gauß (1801) to Szyjewski (2005))

A number of interesting local links that you are warmly encouraged to explore:

• Problem of the Month sponsored by the Stony Brook mathematics deptartment. The first two winners each month get $25!
• Math Club

Math Learning Center

The Math Learning Center (MLC), located in Room S-240A of the Math Tower, is an important resource. It is staffed most days and some evenings by mathematics tutors (professors and advanced students). For more information and a schedule, consult the MLC web site.

Special needs

If you have a physical, psychiatric, medical or learning disability that may impact on your ability to carry out assigned course work, you may contact the Disabled Student Services (DSS) office (Humanities 133, 632-6748/TDD). DSS will review your concerns and determine, with you, what accommodations may be necessary and appropriate. I will take their findings into account in deciding what alterations in course work you require. All information on and documentation of a disability condition should be supplied to me in writing at the earliest possible time AND is strictly confidential. Please act early, since I will not be able to make any retroactive course changes.