MA 1560: Number Theory

Instructor

  • Reinier Bröker
  • Office: Room 117, Math Building
  • Phone: (401) 863-7959
  • E-mail: reinier@math.brown.edu

    • Times and Places

    Announcement

    You have the possibility to earn 7 points on your midterm grade by doing an extra credit problem: Chapter 8.20. Due date is Tuesday 4/19.

    Text book

    A classical introduction to modern number theory by K. Ireland and M. Rosen. This book is available at the Brown bookstore and online. There is both a softcover and a hardcover edition, the contents of these versions should be identical.

    Contents

    A basic introduction to the theory of numbers. Unique factorization, prime numbers, modular arithmetic, quadratic reciprocity, quadratic number fields, finite fields, Diophantine equations, and additional topics.

    Grade

    Your grade is based on the homework, midterm, research project and final. I will weigh these as follows. Homework: 15%, Research Project: 15%, Midterm: 30%, Final: 40%. If any curving is done (it need not be), it will only be done on your averaged grade.

    Midterm

    Tuesday March 22nd, in class, room TBA.

    Research Project

    Part of this course is learning to do literature research and writing mathematics. Details about the project can be found here.

    Final

    The final exam will be Friday May 13, 9am -- noon. The room will be announced later.

    Homework

    I will assign homework every week, see below. The exercises fall into three categories: (1) easy exercises, (2) hard exercises, (3) obligatory exercises. Easy exercises are worth 2 points, hard and obligatory exercises are worth 3 points. Your homework score is your total number of points. If you decide to do mostly easy exercises, then you should expect that you're at `B-level' for the course. The hard exercises are mostly on `A-level'. You have the freedom to choose the exercises you do yourselves. I will tell you how many exercises you should hand in and a set of exercises you may pick from.

    Calculators

    Calculators are not required for this course, and they are not allowed on any exam. You make your own life easier if you try to not use them for the homework exercises either.

    Computers

    It can be convenient to use a computer to do some computations faster. One can use it to compute say the greatest common divisor of two large number. I encourage you to familiarize yourself with one of the computer algebra packages described below. Be mindful though that you should always understand what the computer is doing internally to perform your computation.

    The free program Sage is rapidly gaining popularity. You can download and install it on your own computer, or you can use the web interface after you created a username and password.

    Many researchers rely on Pari-GP for computational work. Pari is free as well, but I personally find it a bit hard to use. Pari is incorporated into Sage.

    Homework

    Due date Number of exercises Easy exercises Hard exercises Obligatory exercises
    2-3-2011 5 Chapter 1: 1, 10, 11, 13, 19 Chapter 1: 6, 7, 9, 14, 15, 16, 17, 18, 20, 21 Compute the gcd of your phone number and birthday, Review exercises (you don't have to hand these in)
    2-10-2011 6 Chapter 1: 24, 27, 28. Chapter 2: 9, 10, 11 Chapter 1: 23, 25, 26, 30. Chapter 2: 1, 2, 4, 5 Chapter 2: 3
    2-17-2011 6 Chapter 2: 15, 16, 17. Chapter 3: 1, 2. Chapter 2: 18, 20, 27. Chapter 3: 10, 17, 19, prove that there are infinitely many primes congruent to 1 modulo 4. Chapter 2: Make the constant c_2 in Prop 2.4.4 explicit
    2-24-2011 6 Chapter 4: 1, 2, 3, 13, 20, 23 Extra: 1,2,3,4,5, Chapter 4: 6, 9, 15, 18, 22 Extra: 1, 2
    3-03-2011 3 Chapter 5: 2 Chapter 5: 3, 4, 14, 15 Chapter 5: 16
    3-10-2011 3 Chapter 6: 1, 2 Chapter 5: 18, 38 Chapter 5: 22 (pick 2 of them), Extra: 1
    3-17-2011 6 Chapter 6: 9, 10, 17, 19. Chapter 7: 1 Chapter 6: 4, 11, 15, 16, 18, 23. Chapter 7: 6, 10, 18
    3-24-2011 3 Chapter 8: 1, 2, 3, 4, 5, 6
    4-7-2011 Project due!
    4-14-2011 4 Ch 8: 7, 11, 12, Ch 8: 13, 14, 15, 16 Prove Lemmas 3,4,5 of Chapter 9.7 using the method from class. (This counts as two exercises.)
    4-19-2011 Extra credit problem (8.20) due!
    4-21-2011 5 Chapter 9: 1, 2, 8, 15 Chapter 9: 14 with D replaced by Z, 16, 23, 24, 25, 26 Chapter 9: 7, 14
    4-28-2011 2 Use cubic reciprocity to check if 14 is a cube modulo 37, Chapter 9: 28
    5-5-2011 1 Extra