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)
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