Catalog Description

MATH 306 Number Theory (3-0-3)(F) Diophantine equations, modular arithmetic, quadratic reciprocity, primality testing and factoring methods.
PREREQ: MATH 175 and MATH 187.

Prerequisites

MATH 175 Calculus and Analytic Geometry and MATH 187 Discrete and Foundational Math, or equivalent courses elsewhere, or permission of the instructor. The rationale for these prerequisites is that Number Theory requires a level of Mathematical maturity which could be attained through these two lower division courses.

Jurisdiction

This course is not currently controlled by a departmental committee and individual instructors may choose different textbooks. Exams, homework, and grading system are left to the instructor.

Learning Objectives

As a course in Mathematics, the objectives of MATH 306 include that students be able to:

use results from elementary number theory to solve contemporary problems;
explain from elementary principles why certain number theoretic facts are true;
use available computer programs to solve problems from the genre of computational number theory.

Currently this course is taken mainly by mathematics majors.

Upon completion of this course, students should be able to:

explain the fundamental concepts of modular arithmetic;
use available software to do the computational part of problem solving;
use the Chinese Remainder Theorem to solve systems of equations in modular arithmetic;
use the Chinese Remainder theorem to compute modular square roots;
use the Quadratic Reciprocity Theorem to determine when a given number has a modular square root.
solve special instances of the Discrete Logarithm Problem;
factor large numbers which are a product of two prime numbers, given the phi-value of the number;
explain and apply Fermat's factoring method, and Pollard's Rho method for factoring.

Assessment of Learning Objectives

Students will be assessed by evaluating their ability to do problems based on the learning objectives. The problems may occur in several contexts:

Periodic problem sets for homework serve both as learning and as assessment tools. Classroom activities may vary depending on students' outcomes on homework assignments.
Problems given on take-home exams are designed to evaluate a student's ability to solve more complicated and time consuming problems. Moreover, take-home problems give students the opportunity to demonstrate their ability to use technology to solve problems involving real data or otherwise not amenable to simple analytic techniques.
Problems given on in-class exams are designed to give students the opportunity to demonstrate their ability to work simpler and less time consuming problems which probably do not involve real data.

Topics and Approximate Timeline

The following table is based on a typical semester schedule-60 class meetings of 50 minutes each. The actual amount of time spent on each topic will vary slightly from semester to semester and instructor to instructor.


M 306 Number Theory
	Number of
Topic	Meetings
Greatest Common Divisor and Euclidean Algorithm	3
Fundamental Theorem of Arithmetic	3
Modular Arithmetic and Linear Equations	3
Modular Arithmetic and Exponentiation	6
The infinitude of the prime numbers	4
Modular Arithmetic and root computation	4
Primitive roots and the Discrete Log Problem	6
Modular Arithmetic and Square roots	6
The Quadratic Reciprocity Theorem	6
Primality Tests	8
Factoring Algorithms	8
Exams	3
Total	60

Text

The current text is A Friendly Introduction to Number Theory, Joseph H. Silverman, Prentice Hall (1997). Another textbook that has been used in recent years is Elementary Number Theory, David M. Burton, McGraw-Hill.

Web sites

The following websites offer valuable learning materials for this course:

http://www.mersenne.org/prime.htm The great internet Mersenne Prime Search
http://www.utm.edu/research/primes/ The Prime Pages: prime number research, records and resources
http://www.entropia.com/primenet/status.shtml Internet PrimeNet Server
http://www.npac.syr.edu/factoring/ The RSA factoring by web project

Format, Student Activities, and Grades

Class meetings involve a combination of lecture, questions and discussion, and sometimes small group activity; the instructor chooses the appropriate mix. The computer algebra system, Maple, is used for laboratory activities and homework. Homework is an important part of the course; many exercises involve extensions of ideas in the text to new situations, rather than just routine applications. Some exams may be partially take-home. The instructor chooses the exact grading scheme, but a typical distribution might be:


Homework (including a writing assignment and project)	33 ¹/₃ %
Exams	33 ¹/₃%
Final Exam	33 ¹/₃%
Total	100%

Letter grades are based on a scale in which 90% of the total possible points guarantees an A, 80% a B, 70% a C, and 60% a D, with the instructor having the discretion to lower these cut-offs if warranted.

File translated from T_EX by T_TH, version 1.56.