MATH 330 – Number Theory – Spring 2014 Instructor: Lucian Ionescu; Contact Info: STV 312J 438-7167;[email protected]; Office Hours: TBA, or by appointment at other times. Textbook: Number Theory: an introduction via the Distribution of Primes, Benjamine Fine and Gerhard Rosenberg, 2007, ISBN-13 978-0-8176-4472-7, Content: Number Theory is a study of number systems from an elementary point of view, but it may “hide” a lot of “higher” math, e.g. group theory (algebraic number theory), analysis (analytic number theory), even geometry! The main topics and problems in this course revolve around prime numbers, congruencies and secret messages; a glimpse into elliptic curves (“arithmetic doughnuts”, as 2D version of trigonometry on the circle) is included as “bonus material”. We will use SAGE (online computer algebra system) for examples and computational explorations. Specifically, you will learn about divisibility, primes, unique factorization, linear congruencies and Diophantine equations, number-theoretic functions, primitive roots, quadratic reciprocity, continued fractions and cryptography. Prerequisites: A grade of C or better in MAT 260. Objectives: The general goals of the course are 1) To apply and develop some of the topics and mathematics techniques of discrete mathematics from MAT 260; 2) To introduce the student to applications of number theory in physics and computer science; 3) To emphasize the connections with the basic algebraic structures, and their geometric significance using graphical representations. In general it is the aim of the course to provide students with an opportunity to enhance their ability to learn, understand and present topics of mathematics, in a beautiful area of mathematics loved by most famous mathematicians (e.g. Gauss) and hobbyists alike (e.g. Fermat). Course Format: You are responsible for reading assignments, and this is not a trivial task. Reading assignments are to be completed before the next scheduled meeting of the class. During the class, we will focus on understanding the concepts and methods involved. After the class, write one page outline of the corresponding section, before attempting the homework. Attempt all the homework problems before the next class period, when you should turn in the outline for one point per submission. Before discussing the new assigned material, we will discuss your difficulties or questions regarding the homework problems. Attendance and active participation are expected. Outlines: After each class, and after studying the assigned material (maybe producing “study notes” aside), review the assigned material and collect the main “mathematical facts” (concepts and their properties, theorems) as a short outline (usually one page long; informally written, to help you organize the information and “digest” the essentials, discarding the “details” at this stage). Homework: will be assigned weekly, in class and posted on the website. Homework is due and will be collected Wednesdays, together with the outlines (one pile; keep them together as one submission). Check the course’s web site for updates. Exams: In class quizzes will replace the usual exams during the semester, and a final exam. Evaluation: Grades will be based on the following points: Homework and Outlines 200 Class activities (Attendance and participation): 100 In-class quizzes 100 Final exam 200 Total 600 points. The grading scale is based on: A [90-100%], B [80-90%), C [70-80%), D [60,70%), F [0,60%).

N.B.: Students who believe they may need accommodations in this class/program are encouraged to contact the Disability Access Center as soon as possible to better ensure that such accommodations are implemented in a timely fashion. “Warning: Plagiarism and cheating are serious offenses. Penalties can range from a minimum of a zero grade on the invalid instrument to expulsion from the University” Topics Covered

The topics are listed below, according to a tentative pace of covering the material. Adjustments will be made as we go.

Week Sections Topics Week Sections Topics 1 Introduction 9 Spring break 1/13 Ch.1 Historical remarks No Classes 2.1 The ring of integers 2 2.2 Divisibility, primes and composites 10 1/20 2.3 The Fundamental Th. of Arithmetic R&p (review & problems) 3 11 1/27

4 12

5 13

6 14

7 15

8 16 Review and overview