Number Theory – Mathematics 0420
Brown University – Spring, 2018
Professor Joseph Silverman

Text A Friendly Introduction to Number Theory  (4th edition, 2012) by Joseph H. Silverman, Pearson, ISBN: 978-0-321-81619-1
     You may freely download Chapters 1–6 and other material on the FRINT home page .
Office Mathematics Department, Kassar House, Room 202
Phone 863-1124
Email jhs@math.brown.edu
Web Site www.math.brown.edu/~jhs/MA0042/MA0042HomePage.html
Office Hours Monday 2:00–2:45pm and Tuesday 10:00–11:00am
The last regularly scheduled office hour will be Tuesday, May 8, 10:00–11:00am.
There will be one additional extra office hour on Tuesday, May 15, 3:00–4:00pm.
Click here for Information about Office Hours and Other Academic Resources
Course Time TuTh 2:30–3:50 PM (K Hour)
Course Location Barus Holley 161
Homework I will keep a list of reading and homework assignments on the following web page:
Click here to go to the Math 0420 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. (If you haven't finished the HW, just hand in what you've done. A few missing problems during the semester are unlikely 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 0420 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 compute the greatest common divisor of two large numbers, but you should be sure that you understand how the computer is doing the computation.
Computer Package for Math 0420 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. If you're computer savvy, you can download PARI here:
Click here to go to the PARI home page.

Another way to use PARI to do short calculations is to use the SAGE web site. Or you can use SAGE itself, which is another computer algebra package that can be used for number theory computations.

As an alternative, I have written a web page number theory calculator that you can use for Math 0420. It is not as versatile as PARI or SAGE 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 for an online number theory calculator.

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

Hour Exam #1

Tuesday February 27

Covers Ch. 1, 2, 4–10 

Exam 1 Solutions

Hour Exam #2

Tuesday April 10

Covers Ch. 11–12, 14–18, 20–22

Exam 2 Solutions

Final Exam

Wednesday May 16
Exam Group 11

Time 2:00-4:00pm
BH 153

The final exam is cumulative. It covers
Chapters 1-2, 4-12, 14-18, 20-22, 24-25, 31-34, 37-39
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

  1. Pythagorean Triples
  2. Divisibility, greatest common divisors, and linear equations
  3. The fundamental theorem of arithmetic
  4. Congruences; Fermat's little theorem ap-1≡1 (mod p); Euler's formula aφ(m)≡1 (mod m)
  5. The Chinese remainder theorem
  6. Primes, Mersenne primes, and perfect numbers
  7. Powers, roots, and the RSA public key cryptosystem
  8. Squares modulo p and quadratic reciprocity
Additional Topics Chosen From: (topics covered during Spring 2018 marked with a *)
  1. Sums of two squares*
  2. Fibonacci numbers and linear recurrences*
  3. Pell's equation and Diophantine approximation*
  4. Binomial coefficients*
  5. Continued fractions and Pell's equation
  6. Sums of powers
  7. Elliptic curves and Fermat's last theorem
  8. Primitive roots
  9. Irrational and transcendental numbers*
  10. Descent and solutions to X4+Y4=Z4

Go to Professor Silverman's Home Page.