Elementary Number Theory: Lessons

• Introduction to Number Theory: Course Structure
• Introduction to Number Theory: Study of Z
• Famous Theorems about Primes
• Pythagorean Triples, Diophantine Equations, Fermat's Last Theorem
• The Euclidean Algorithm: Introduction and Statement
• The Euclidean Algorithm: Proof that it works
• Relatively Prime Integers
• The Fundamental Theorem of Arithmetic (Unique Factorization)
• Consequences of Unique Factorization
• Examples of Multiplicative Functions
• Multiplicative Functions; Perfect Numbers
• Linear Diophantine Equations
• Introductory Remarks about Congruences
• Modular Arithmetic
• Residues
• Linear Congruence Equations
• Chinese Remainder Theorem
• Linear Congruence Equations in Two Variables
• Reduced Residue System; Euler's Phi Function
• Euler's Theorem, Fermat's Little Theorem, Pseudoprimes
• A Formula for Euler's Phi Function
• Polynomial Congruence
• Roots of Polynomials Mod n
• Degree of a Polynomial, Factoring, Wilson's Theorem
• Definition of Order Mod n of a; k-th Roots; Primitive Roots
• Roots and Order
• Counting Roots; Square Roots of -1
• Magic Squares (Intro)
• Filling Squares
• Producing Magic Squares
• The Lemma Behind Uniform Step Method Magic Squares
• Diabolic Squares
• Introduction to Diophantine Equations
• Using Congruences to Solve Diophantine Equations
• Pythagorean Triples
• The Method of Descent (Fermat)
• The Method of Ascent
• Decimal Expansions of Rational Numbers
• Rational and Irrational Numbers
• Continued Fractions; Geometric Preliminaries
• The Continued Fraction Algorithm
• The Continued Fraction Approximation; Estimating Closeness
• Computing the Integral Coefficients a (sub n)
• Introduction to Quadratic Fields
• Defining Equations and Quadratic Integers
• Characterizing Quadratic Integers
• Factorization into Quadratic Primes
• Unique Factorization
• More on Unique Factorization Domains
• Euclidean Fields
• More on Euclidean Fields and UFDs
• Integers which are Sums of 2 Squares
• Using Maple to Help Prove a Theorem (4 and 5 Squares)