Number Theory

Number Theory Course Menu

Topics include: distribution of primes, representations of integers, Fibonacci numbers, divisibility, Euclidean algorithm, fundamental theorem of arithmetic, number-theoretic functions, Diophantine equations, congruence, primitive roots, the Chinese remainder theorem, quadratic residues, and elementary partition theory.
Prerequisite: Methods of Proof and Linear Algebra with a grade of C or better.

Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor.

Chapter 1

Basis Representation

Lessons Homework
1.1   The Principles of Mathematics Induction 1.1
1.2   The Basis Representation Theorem 1.2

Chapter 2

The Fundamental Theorem of Arithmetic

Lessons Homework
2.1   Euclid's Division Algorithm 2.1
2.2   Divisibility 2.2
2.3   The Linear Diophantine Equation 2.3
2.4   The Fundamental Theorem of Arithmetic 2.4

Chapter 3

Combinatorial and Computational Number Theory

Lessons Homework
3.1   Permuations and Combinations 3.1
3.2   Fermat's Little Theorem 3.2
3.3   Wilson's Theorem 3.3
3.4   Generating Functions 3.4
3.5   The Use of Computers in Number Theory 3.5

Chapter 4

Fundamentals of Congruences

Lessons Homework
4.1   Basic Properties of Congruences 4.1
4.2   Residue Systems 4.2
4.3   Riffling 4.3

Chapter 5

Solving Congruences

Lessons Homework
5.1   Linear Congruences 5.1
5.2   The Theorems of Fermat and Wilson Revisited 5.2
5.3   The Chinese Remainder Theorem 5.3
5.4   Polynomial Congruences 5.4

Chapter 6

Arithmetic Functions

Lessons Homework
6.1   Combinatorial Study of phi(n) 6.1
6.2   Formulae for d(n) and sigma(n) 6.2
6.3   Multiplicative Arithmetic Functions 6.3
6.4   The Mobius Inversion Formula 6.4

Chapter 7

Primitive Roots

Lessons Homework
7.1   Properties of the Residue System 7.1
7.2   Primitive Roots Modulo p 7.2

Chapter 8

Prime Numbers

Lessons Homework
8.1   Elementary Properties of pi(x) 8.1
8.2   Tchebychev's Theorem 8.2
8.3   Some Unsolved Problems About Primes 8.3

Chapter 9

Quadratic Residues

Lessons Homework
9.1   Euler's Criterion 9.1
9.2   The Legendre Symbol 9.2
9.3   The Quadratic Reciprocity Law 9.3
9.4   Applications of the Quadratic Reciprocity Law 9.4

Chapter 11

Sums of Squares

Lessons Homework
11.1   Sum of Two Squares 11.1
11.2   Sum of Four Squares 11.2

Chapter 12

Elementary Partition Theory

Lessons Homework
12.1   Introduction 12.1
12.2   Graphical Representation 12.2
12.3   Euler's Partition Theorem 12.3
12.4   Searching for Partition Identities 12.4

Chapter 13

Partition Generating Functions

Lessons Homework
13.1   Infinite Products as Generating Functions 13.1
13.2   Identities Between Infinite Series and Products 13.2
13.3   Euler's Partition Theorem 13.3
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