This course on Number Theory aims to teach students the principles and concepts related to divisibility, linear Diophantine equations, congruences, and quadratic reciprocity. The course covers topics such as the Euclidean Algorithm, Fermat's Little Theorem, the Chinese Remainder Theorem, and the Mobius Inversion Formula. Students will also learn about primitive roots modulo p, the Legendre Symbol, and applications of quadratic reciprocity. The teaching method includes video lectures originally created for a specific semester. This course is intended for individuals interested in advanced mathematics and number theory.
Overview
Syllabus
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.
Taught by
The Math Repository
