Number Theory – Mathematics 1560
Brown University – Spring, 2010
Professor Joseph Silverman

Text A Classical Introduction to Modern Number Theory
by Kenneth Ireland and Michael Rosen
Springer-Verlag
ISBN 978-0387973296
Note to students: Do not be alarmed that the textbook is part of Springer's "Graduate Texts in Mathematics" series. The first half of the book was written by Ireland and Rosen specifically to be used in Math 1560.
Office Mathematics Department, Kassar House, Room 202
Phone 863-1124
Email jhs@math.brown.edu
Web Site www.math.brown.edu/~jhs/MA0156/MA0156HomePage.html
Office Hours TBA (Or send me an email to make an appointment. I tend to be in on MWF and not on TTh.)
Course Time MWF 10:00–10:50am (C hour)
Course Location TBA
Problem Sets NOTE: The problem sets are challenging. Don't leave them until the last minute! We will be moving rapidly. In order to learn the material, it is very important to DO THE HOMEWORK WHEN IT IS ASSIGNED.
RULES: Homework must be stapled. All problems must be clearly labeled. Late homework will not be accepted under any circumstances. (One or two missing homeworks won't affect your grade too much, and it's an imposition on the grader to have to go back and grade late homeworks.)
Note on Using Computers in Math 156 Computers are a useful tool that can be used to generate data for making conjectures and to perform computations that would be tedious to do by hand. However, they are not a replacement for understanding. So for example, one might use a computer to calculate the greatest common divisor of two large numbers, but you should be sure that you understand how the computer is doing the computation.
Homework on the Web The homework assignments will be posted on the web. (I'll also try to announce them in class.)
Click here to go to the Math 156 Web Homework Page.

Dates to Remember: There will be an in-class midterm exam and a final exam.

Midterm

Date TBA

In class

Final Exam

Date 05/17/2010
Exam Group 3

Time 9:00am - Noon
Location TBA

Grading: The course grade will be determined on the following basis:

Problem Sets

20%

Additional Assignment(s) TBA

10%

Midterm

25%

Final Exam

45%

Tentative Syllabus:

  1. Divisibility
    • Greatest common divisor and the Euclidean algorithm
    • Fundamental Theorem of Arithmetic (for Z, Z[i], and F[T])
  2. Congruences
    • Solution of linear congruences
    • Chinese remainder theorem
    • Fermat's little theorem: ap-1=1 (mod p)
  3. Arithmetic functions
    • Euler's φ function, number of divisors d(n), sum of divisors σ(n)
    • Euler's formula: a&phi(m)=1 (mod m)
    • Multiplicative functions, Dirichlet product, Moebius inversion formula
  4. Prime numbers
    • Infinitude of primes (with congruence conditions)
    • Estimates for π(x)
    • Mersenne primes and application to perfect numbers
  5. Finite fields and primitive element theorem for Fq*
  6. Quadratic residues
    • Legendre and Jacobi symbols
    • Euler's criterion a(p-1)/2=(a|p) (mod p) and Gauss' criterion (-1)μ=(a|p)
    • Proof of quadratic reciprocity

Additional Topics Chosen from:

  1. Solving congruences modulo pn, Hensel's lemma, and the ring of p-adic numbers Zp
  2. Pell's equation x2-Dy2=1 and units in real quadratic fields
  3. Riemann ζ function: Euler product, analytic continuation, special values ζ(2k), and Bernoulli numbers
  4. Elliptic curves (over Q and/or over Fp)
  5. Diophantine equations over Fp and over Fq (upper bounds, zeta functions, Weil conjectures)
  6. Average values of arithmetic functions

Go to Professor Silverman's Home Page.