THEORY OF NUMBERS
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
a. Definition and Elementary Properties of Congruences
b. Residue Classes, Reduced Residue Systems, and Euler's Function
c. Solution of Congruences
- The Powers of an Integer, Modulo m
a. The Order of an Integer (Mod. m)
b. Integers Belonging to a Given Exponent (Mod. m)
- 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.
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