MA483 INTRODUCTION TO NUMBER THEORY (3 Cr.)

COURSE DESCRIPTION

Prerequisite: MA 312 (Abstract Algebra with Applications) or permission of instructor

Course Description
Peano's axioms, Euclidean algorithm, congruence, quadratic reciprocity law, Gaussian integers, continued fractions, Diophantine equations, and theory of equations.

Goal/Purpose
The purpose of MA483 is to study the properties of the integers and various finite integer systems.  The material presented will be "classical", yet numerous modern applications of the ideas will be given.

Course Content Outline

1. Divisibility Theory of the Integers
   1. Division algorithm
   2. Greatest Common divisors and Euclid's algorithm
   3. Solving AX + BY = C in the integers
   4. Applications
      1. gears
      2. symbolic fractions and their arithmetic on computers
2. Primes
   1. Fundamental theorem of arithmetic
   2. Sieve of Erathosthenes
   3. Application (Data Security Systems)
3. Theory of Congruences
   1. Modular arithmetic
   2. Solving linear equations
   3. Fermat's little theorem
   4. Wilson's theorem
   5. Applications
      1. calendar problems
      2. computer applications (circular queues, etc.)
4. Number Theoretic Functions
   1. Examples (Euler's phi-function, etc.)
   2. Mobius inversion formula
5. Solving Quadratic Equations
   1. Order of an integer modulo N
   2. Primitive roots
   3. Diophantine equations
   4. Fibonacci numbers
   5. Applications

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