Number Theory

Dive into the fundamentals of number theory through a comprehensive video lecture series originally created for a Fall 2017 course. Explore topics ranging from mathematical induction and the division algorithm to quadratic reciprocity and Dirichlet's divisor problem. Learn about key concepts such as the Euclidean algorithm, Fermat's Little Theorem, linear congruences, and the Chinese Remainder Theorem. Develop a deep understanding of prime numbers, multiplicative functions, and the Euler phi function. Engage with advanced subjects like Gauss' Circle Problem and sums of squares. Perfect your skills in computer programming applications within number theory and gain insights into the fascinating world of mathematical patterns and relationships.

0 Introduction.
1 1 The Principle of Mathematical Induction.
1 2 The Basis Representation Theorem.
2 1 The Division Algorithm.
2 2a Divisibility.
2 2b The Euclidean Algorithm.
2 3 Linear Diophantine Equations.
2 4 The Fundamental Theorem of Arithmetic.
3 1 Permutations and Combinations.
3 2 Fermat's Little Theorem.
3 3 Wilson's Theorem.
3 5 Computer Programming.
4 1 Basic Properties of Congruences.
4 2 Residue Systems.
5 1 Linear Congruences.
5 2 Fermat's Little Theorem and Wilson's Theorem.
5 3 The Chinese Remainder Theorem.
6 1a The Euler Phi Function Part 1.
6 1b The Euler Phi Function Part 2.
6 2 6 3 Multiplicative Functions.
6 4 The Mobius Inversion Formula.
7 1 Orders of Elements.
7 2 Primitive Roots Modulo p.
8 1 The Prime Counting Function.
9 1 Euler's Criterion.
9 2 The Legendre Symbol.
9 3a Quadratic Reciprocity Part 1.
9 3b Quadratic Reciprocity Part 2.
9 4 Applications of Quadratic Reciprocity.
10 1 Consecutive Residues.
10 2a Consecutive Triples of Residues Part 1.
10 2b Consecutive Triples of Residues Part 2.
11 1 Sums of Two Squares.
11 2 Sums of Four Squares.
15 1 Gauss' Circle Problem.
15 2 Dirichlet's Divisor Problem.
Infinity Conclusion.