COURSE NUMBER AND TITLE: MATH 4320 Theory of Numbers

CREDIT HOURS: 3

CATALOG DESCRIPTION: A study of the positive integers including divisibility, prime numbers and the theory of congruences. Additional topics may include Fermat’s theorem, the law of quadratic reciprocity, and perfect numbers.

PREREQUISITE(S): MATH 2012 or MATH 2030

SUGGESTED TEXT(S):
Number Theory by George Andrews
Elementary Number Theory by David Burton
Number Theory and its History by Oystein Ore

COURSE OUTLINE:

1. Basic concepts
   • Divisibility in the integers and prime integers
   • The greatest common divisor and the Euclidean Algorithm
   • The Fundamental Theorem of Arithmetic
   • Modular arithmetic and congruences
   • Divisibility tests
   • Linear congruences (and study of the units mod n)
   • Diophantine equations
2. More advanced concepts
   • The Chinese Remainder Theorem
   • Fermat's Little Thoerem and Wilson's Theorem
   • Euler's phi-function
   • Euler's Theorem
3. Other topics
   • Primality tests and pseudoprimes
   • Factoring algorithms and Fermat's method
   • Public-key cryptography
   • The functions mu, tau, and sigma
   • Multiplicative functions and Mobius inversion
   • Quadratic reciprocity and the Legendre symbol
   • Continued fractions