MATH 412

THEORY OF NUMBERS

Catalog Entry

MATH 412. Theory of Numbers

Three hours lecture (3).

Prerequisite: MATH 300

Study of divisibility, primes, congruences, diophantine equations and quadratic residues.

Detailed Description of Content of Course

- Introductory Concepts

a. Nature of Number Theory

b. Methods of Proof

c. Radix Representation - The Euclidean Algorithm and its Consequences

a. Divisibility, Greatest Common Divisor, and Least Common Multiple

b. The Euclidean Algorithm

c. The Fundamental Theorem of Arithmetic

d. The Linear Diophantine Equation - Congruences

a. Definition and Elementary Properties of Congruences

b. Residue Classes, Reduced Residue Systems, and Euler's Function

c. Solution of Congruences

i. Linear

ii. Polynomial

iii. Quadratic - The Powers of an Integer, Modulo m

a. The Order of an Integer (Mod. m)

b. Integers Belonging to a Given Exponent (Mod. m)

c. Indices - Continued Fractions

a. Basic Identities

b. The Simple Continued Fraction Expansion of a Rational Number

c. The Expansion of an Irrational Number - The Gaussian Integers

a. Divisibility, Units, and Primes

b. The Greatest Common Divisor

c. The Unique Factorization Theorem - Diophantine Equations

a. The Equations x 2+y 2=z 2 and x 4+y 4=z 4

b. The Equations x 2-dy 2=1 and x 2-dy 2= -1

c. Dell's Equation

Applications and the history of number theory will be discussed as appropriate throughout the course.

Detailed Description of Conduct of Course

Most instructors use the lecture-discussion method. Some may require students to work together in small groups. Students may be required to work problems on the chalkboard.

Goals and Objectives of the Course

Students are expected to gain knowledge of, and skills with, the basic theorems of number theory.

Assessment Measures

Graded tasks may include tests, quizzes, homework exercises, class participation, and attendance.

Other Course Information

This course is intended as an elective for majors and minors in mathematics

Review and Approval

DATE ACTION APPROVED BY

Sept. 2001 Review Stephen Corwin, Chair