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.

Basis Representation

Lessons | Homework |
---|---|

1.1 The Principles of Mathematics Induction |
1.1 |

1.2 The Basis Representation Theorem |
1.2 |

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 |

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 |

Fundamentals of Congruences

Lessons | Homework |
---|---|

4.1 Basic Properties of Congruences |
4.1 |

4.2 Residue Systems |
4.2 |

4.3 Riffling |
4.3 |

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 |

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 |

Primitive Roots

Lessons | Homework |
---|---|

7.1 Properties of the Residue System |
7.1 |

7.2 Primitive Roots Modulo p |
7.2 |

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 |

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 |

Sums of Squares

Lessons | Homework |
---|---|

11.1 Sum of Two Squares |
11.1 |

11.2 Sum of Four Squares |
11.2 |

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 |

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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