Number Theory – Mathematics 156
Brown University – Spring, 2006
Professor Joseph Silverman

Text A Classical Introduction to Modern Number Theory (2nd edition)
by Kenneth Ireland and Michael Rosen
Springer-Verlag GTM Series, ISBN 038797329X
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 Reading and Exam Period Office Hours:
Weds, May 3, 10:00 - 11:00
Fri, May 5, 11:00 - 12:00
Mon, May 8, 9:00 - 10:00 and 3:00 - 4:00
Weds, May 10, 10:00 - 11:00 and 3:00 - 4:00
Course Time MWF 9:00 - 9:50 AM (B hour)
Course Location Barus Holley 153
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 isn't going to affect your grade, 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, we'll use computers to compute the greatest common divisor of large numbers, but you should be sure that you understand how the computer is doing the computation.
Computer Package for Math 156 In order to eliminate some of the tedium of numerical calculations, we will be using the computer program PARI-GP. I will spend part of one class explaining how to use PARI. The good news is that PARI is free, and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly. You may find it most convenient to have a copy of PARI onto your own computer, but I'll also create a Math 156 guest account on the Math Dept's system. If you are running Windows, you can directly download an executable file. If you're running Mac OS X or Linux (or some other version of Unix), you can download source files and compile them.
Click here to go to the PARI home page.
Another way to use PARI to do short calculations is on the following web site. Type one or more lines in the top box and hit the PARI button. The output appears in the bottom box.
Click here for an online PARI calculator.
Homework on the Web Some of the homework problems may be posted on the web. I'll announce them in class and you can download them on the following page:
Click here to go to the Math 156 Web Homework Page.

Dates to Remember: There will be two in-class hour exams and a final exam.

Hour Exam #1

Friday, March 3

In class

 Click to download Exam 1 Solutions.

Hour Exam #2

Friday, April 7

In class

 Click to download Exam 2 Solutions.

Final Exam

Thursday, May 11
Exam Group 02

9:00am-12:00am
Barus-Holley 157

 Click to download Final Exam Solutions.

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

Problem Sets

20%

Hour Exams (22.5% each)

45%

Final Exam

35%

Syllabus (Primary Topics):

  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)
  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
  7. Diophantine equations
    • Linear equations ax+by=c
    • Pell's equation x2-Dy2=1 and units in real quadratic fields
    • Fermat's equation x4+y4=z4 and descent
  8. Analytic number theory
    • Riemann ζ function, Euler product, analytic continuation (at least to s > 0)
    • Characters and L-series

Syllabus (Additional Topics as Time Permits):

  1. Cryptography (RSA and/or discrete logarithm based systems)
  2. Elliptic curves (over Q and/or over Fp)
  3. Diophantine equations over Fp and over Fq (upper bounds, zeta functions, Weil conjectures)
  4. Solving congruences modulo pn and Hensel's lemma
  5. Partitions
  6. Computational number theory: Primality testing, factorization algorithms, discrete logarithm algorithms
  7. Special values ζ(2k) and Bernoulli numbers
  8. Average values of arithmetic functions

Go to Professor Silverman's Home Page.