Number Theory

This is indeed a nice textbook covering a number of elementary topics in number theory. The book includes a good deal of numerical examples, which are analyzed for patterns and used to make "conjectures". Various other chapters provide brief but insightful and motivating excursions into topics like Mersenne Primes, number sieves, RSA cryptography, elliptic curves, etc. There are many other excellent undergraduate books on the subject. Here is a sample (all of them available in our library):

*Elementary Number Theory and Its Applications*, by Kenneth Rosen, (fourth edition)*An Introduction to the Number Theory*, H.M. Stark*Number Theory*, G.E. Andrews*Introduction to Analytic Number Theory*, T.M. Apostol*Lectures on Number Theory*, P.G.L. Dirichlet with supplements by R. Dedekind*The higher arithmetic*, H. Davenport*An Introduction to the Theory of Numbers*, I. Niven and H.S. Zuckerman*A Classical Introduction to Modern Number Theory*, K. Ireland and M. Rosen*Fundamentals of Number Theory*, W.J. LeVeque*Number theory with computer applications*, R. Kumanduri and C. Romero

Topic | Sections in textbook | Week | Notes |
---|---|---|---|

Overview / Introduction / Review | Chapter 1 | 1/26-1/30 | |

Pythagorean Triples / Sums of Higher Powers | Chapters 2,3,4 | 2/2-2/6 | |

Divisibility, Euclidean algorithm, Fundamental theorem of arithmetic | Chapters 5, 7 | 2/9-2/13 | |

Linear Diophantine equations / Congruences | Chapters 6, 8 | 2/16-2/20 | |

Fermat's little theorem / Euler's Formula | Chapter 9,10 | 2/23-2/27 | |

Multiplicative functions | Chapter 11, 19 | 3/1-3/5 | First project due 3/4 |

Prime numbers | Chapter 12,13,14 | 3/8-3/12 | Midterm 3/11 |

Powers modulo | Chapter 16,17 | 3/15-3/19 | |

Public key cryptography | Chapter 18 | 3/22-3/26 | |

Primitive roots | Chapter 20, 21 | 3/29-4/2 | |

Quadratic residues | Chapter 22, 21, 23, 24 | 4/12-4/16 | |

Sums of squares | Chapter 25, 26 | 4/19-4/23 | Second project due 4/22 |

Primality testing | Chapter 32 | 4/26-4/30 | |

Cubic curves and elliptic curves | Chapter 40 | 5/3-5/7 | Final exam 5/18, 2-4:30 |

Your grade will be based on the weekly homeworks (20%), two projects (15% each), midterm (20%), and the final exam (30%). The two lowest homework grades will be dropped before calculating the average.

- An On-Line Encyclopedia of Integer Sequences.
- Fibonacci Numbers and Nature. Or Tony Phillips' "The most irrational number". Also "Who was Fibonacci?": a brief biography of Fibonacci.
- Primes: Lots of interesting facts about prime numbers.
- Mersenne Primes: interesting facts about Mersenne primes, perfect numbers, and related topics.
- Primes is P: about a recent polynomial time deterministic algorithm to test if an input number is prime or not.
- RSA: The RSA company's web page containing lots of interesting information about the RSA public key cryptosystem and cryptography in general, from both a technological and a socio-political viewpoint.

- Problem of the Month sponsored by the Stony Brook mathematics deptartment. The first two winners each month get $25!
- Math Club