Number Theory

This course introduces core concepts and results in number theory, including divisibility, Diophantine equations, congruences, quadratic reciprocity, prime counting, sums of squares, and related problems. It also includes a section on computer programming.

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.