MAT 311 Number Theory

Instructor    Sorin Popescu   (office: Math 4-119, tel. 632-8358, e-mail sorin@math.sunysb.edu)

Time and Place    TuTh 02:20pm-03:40pm, SBU 226

 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)

 Elementary Number Theory and Its Applications , by Kenneth Rosen, (fourth edition) Addison-Wesley 2002. This is a very nice textbook that integrates classical (elementary) topics in number theory with lots and lots of applications to cryptology, computer science, etc. It also features a number of computer (programming) projects, mainly for Mathematica and Maple. There are many other excellent undergraduate books on the subject. Here is a sample (all of them available in our library): A Friendly Introduction to Number Theory, J.H. Silverman An Introduction to the Number Theory, H.M. Stark Number Theory, G.E. Andrews Introduction to Analytic Number Theory, T.M. Apostol Lectures on Number Theory, P.G.L. Dirichlet with supplements by R. Dedekind The higher arithmetic, H. Davenport 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, Silverman, Kumanduri and Romero) and more advanced (for example, 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 & Homework assignments

We will cover only part of the textbook and the following schedule may/will be adjusted based on students' preparation and progress. Problems marked with an asterisk (*) are for extra credit.

 Date Topic Homework Notes Wk 1 1/23 1.1 Numbers, sequences, and sums p14/2,4,26,27; p22/5,10,16,30; p28/4,8,22; due 02/04; solutions Wk 2 1/28 1.2 Mathematical induction; 1.3 Fibonacci numbers Fibonacci links 1/30 1.4 Divisibility; 3.1 Prime numbers p34/4,16,28; p76/2,6,10; p84/6,7,13,31; due 02/11; solutions Half hour pretest Wk 3 2/4 3.2 Greatest common divisor Primes links 2/6 3.3-3.4 Euclidean algorithm, Fundamental theorem of arithmetic p94/5,7,19*; p123/2, 3, 6 p104/4, 10, 16*, 32, 35, 46; due 02/18; solutions Wk 4 2/11 3.6 Linear Diophantine equations 2/13 4.1-4.5: Congruences p123/4, 21; p135/5, 22, 26, 28, 38*; p141/2, 6, 8; due 02/25; solutions Wk 5 2/18 p149/4a)-b), 12, 22, 24;p159/1, 10; p167/2, 4, 8b, 14* due 03/04; solutions 2/20 p177/12,19,22; p195/8,12,13,16,17 due 03/11; solutions Wk 6 2/25 2/27 5.1 Divisibility tests Wk 7 3/4 6.1-6.2 Wilson's theorem, Fermat's little theorem, pseudoprimes p202/3, 12, 15, 20, 22, 23; p213/2, 7 due 03/27; solutions First project due 3/6 Wk 8 3/11 Midterm [exam] [solutions] 3/13 6.3 Euler's theorem Recess Wk 9 3/25 7. Multiplicative functions p218/1, 6, 8, 12; p227/1, 2(c,e), 3, 5, 14, 35 due 04/3; solutions 3/27 Wk 10 4/1 p235/2(a-c), 21, 22, 23, 24, 34*, 37* p257/1(a,b), 15,17,18,23; due 04/10; solutions 4/3 8. Cryptography Wk 11 4/8 p267/3, 14, 15 p278/1, 3, 4*, 13, 18, 19 p290/1, 3, 4*, 6, 7, 11*; due 04/22; solutions 4/10 Wk 12 4/15 p304/1, 6, 10; p313/1, 6, 10, 18*; due 04/29; solutions 04/16-04/18 no classes Passover 9. Primitive roots Wk 13 4/22 p319/3, 8, 12, 16; p337/2, 4, 9 due 05/06 Second project due 4/24 Wk 14 4/29 11. Quadratic residues 5/1 Wk 15 5/6 5/8 Review 5/20 Final exam 2:00-4:30pm (SBU 226) 5/20 Review 05/16, 4:00pm-5:30pm (Math Towers P-131)

Homework (see above) and projects (TBA) are an integral part of the course. Problems marked with an asterisk (*) are for extra credit. In addition 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.

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.

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