MAT 311 Number Theory

Instructor:

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

Luis E Lopez (office: Math 2-122, e-mail llopez at math.sunysb.edu)

Schedule:

TuTh 02:20pm-03:40pm, Chemistry 126
Review session: Friday, May 14, 3-4:30pm in Math P-131
Final exam: Tuesday, May 18, 2-4:30pm in Physics P-117

Prerequisites:

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

Textbook(s):

A Friendly Introduction to Number Theory, J.H. Silverman, (second edition), Prentice Hall.

This is indeed a nice textbook covering a number of elementary topics in number theory. The book includes a good deal of numerical examples, which are analyzed for patterns and used to make "conjectures". Various other chapters provide brief but insightful and motivating excursions into topics like Mersenne Primes, number sieves, RSA cryptography, elliptic curves, etc. There are many other excellent undergraduate books on the subject. Here is a sample (all of them available in our library):

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 analytic approaches (for example, Apostol). This course will concentrate only on elementary algebraic number theory, and applications.

Course description:

We will cover only part of the textbook and the following schedule may/will be adjusted based on students' preparation and progress.

TopicSections in textbookWeekNotes
Overview / Introduction / ReviewChapter 11/26-1/30
Pythagorean Triples / Sums of Higher PowersChapters 2,3,42/2-2/6
Divisibility, Euclidean algorithm, Fundamental theorem of arithmeticChapters 5, 72/9-2/13
Linear Diophantine equations / CongruencesChapters 6, 82/16-2/20
Fermat's little theorem / Euler's FormulaChapter 9,102/23-2/27
Multiplicative functionsChapter 11, 193/1-3/5First project due 3/4
Prime numbersChapter 12,13,143/8-3/12Midterm 3/11
Powers modulo mChapter 16,173/15-3/19
Public key cryptographyChapter 183/22-3/26
Primitive rootsChapter 20, 213/29-4/2
Quadratic residuesChapter 22, 21, 23, 244/12-4/16
Sums of squaresChapter 25, 264/19-4/23Second project due 4/22
Primality testingChapter 324/26-4/30
Cubic curves and elliptic curvesChapter 405/3-5/7Final exam 5/18, 2-4:30

Homework and projects (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 2 research/scholarship/computing projects. 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%), two projects (15% each), midterm (20%), and the final exam (30%). The two lowest homework grades will be dropped before calculating the average.