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 Monday 1:30–2:30pm and Friday 9:00–9:50am.
(Or send me an email to make an appointment. I tend to be in on MWF and not on TTh.)

Reading Period and Exam Period Office Hours
No class and no office hours Mon May 3 – Fri May 7
Review Session: Mon May 10, 10:00–10:50am, Wilson 103
Office Hour: Mon May 10, 1:30–2:30pm, Kassar 202
Office Hour: Thurs May 13, 11:00–noon, Kassar 202
Course Time MWF 10:00–10:50am (C hour)
Course Location Wilson Hall 103
Homework Homework assignments will be posted online. (I'll also try to announce them in class.)
Click here to go to the Math 156 Homework Page.
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, you 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.

I tend to use a computer program called PARI-GP to do number theory calculations. The good news about PARI is that it is free and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly. You can download PARI by clicking here. Another way to use PARI to do short calculations is to use the SAGE web site. You'll need to create a (free) account. Then you'll be able to type one or more PARI commands and type Shift-Return to perform the computation.

As an alternative, I have written a web-based number theory calculator that you can use for Math 1560. It is not as versatile as PARI, and it can only handle numbers up to about 16 digits, (and it does not warn you when the numbers get too big, it just gives the wrong answer). However, it it is very easy to use.
Click here to use the online number theory calculator.

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

Midterm

Wednesday March 10

In class — Click to download solutions.

Research Project

Friday March 12 – Friday March 17

Take-home (click to download)

Final Exam

Monday, May 17
Exam Group 3

Time 9:00am - Noon
Barus-Holley 159

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

Problem Sets

20%

Research Project

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.