Tom Roby's Math 3240 Home Page (Fall 2010)
Number Theory
Questions or Comments?
Class Information
COORDINATES: Classes meet Tuesdays and Thursdays
2:003:15 in MSB
215. The registrar calls this Section 001, #14260.
PREREQUISITES: Math 2710 (Transition to
Higher Math)
TEXT:
Number
Theory: A Lively Introduction with Proofs, Applications, and Stories,
1st Edition
by James Pommersheim, Tim Marks,
and Erica Flapan
(Feb. 2010), ISBN=9780470424131.
WEB RESOURCES: The homepage for this course is
http://www.math.uconn.edu/~troby/Math3240F10/.
SOFTWARE In trying to understand properties of integers, we
will often want to generate some data. Doing some computations by
hand is generally good for learning, but having software that can do
bigger computations or check your works is very useful. One free
source on the web
is WolframAlpha. For a
fullfledged progamming environment, check out the free opensource
computer algebra system
called Sage.
GRADING: Your grade will be based on one midterm exam, one
final exam, homework, &
quizzes.
The breakdown of points is:
HW  Quizzes  Midterm  Final


20%  20%  25%  35%


MIDTERM EXAM: Will cover all the material to that point in
the term. It is currently scheduled for TUESDAY 26 October.
Please let me know immediately if you have a conflict with that date.
There are no makeup exams.
QUIZZES: Quizzes will be given roughly once every two
weeks, on the weeks when HW isn't due.
HOMEWORK: Homework will be done in groups of
roughly four students, with only one set of solutions handed in per
group. I've started a list
of HW Policies here,
currently a list of seven.
Many of the assignments will reference Handouts available
here, though I've also put some direct links in below:
Here are the assignments:
 hw1.pdf (#4(d) and due date revised)
 hw2.pdf
 hw3.pdf (3(a) revised)
 hw4.pdf (due date corrected to
10/22; F_5 corrected in 3d)
 hw5.pdf (1(b) group initials corrected
 hw6.pdf
You may find some homework problems to be
challenging, leading you to spend lots of time working on them and
sometimes get frustrated. This is natural. I encourage you to work
with other people in person or electronically. The
HuskyCT site for this class has
discussions boards you can use for this purpose (though I may not check
them regularly). It's OK to get significant help from any resource, but
in the end, please write your own solution in your own words, even if
someone else in your group is the scribe for a given problem.
Copying someone else's work without credit is plagiarism and will be
dealt with according
to university
policy. It also is a poor learning strategy.
CONTENT: Number Theory is a fascinating
subject. It's richness and beauty has captured the imagination of the
greatest mathematicians from antiquity to the current day. Once
thought to be some of the purest (read "most useless") branch of
mathematics, it is now one of the most important: Many of the most
important cryptographic systems, including some crucial to everyday
web commerce, are based on deep unsolved problems in number theory.
DISABILITIES: If you have a documented
disability and wish to discuss academic accommodations, or if you would
need assistance in the event of an emergency, please contact me as soon
as possible.
LEARNING: The only way to learn
mathematics is by doing it! Complete each assignment to the best of
your ability, and get help when you are confused. Come to class
prepared with questions. Don't hesitate to seek help from other
students. Sometimes the point of view of someone who has just figured
something out can be the most helpful.
We will sometimes spend classtime doing things in
groups, presenting mathematics to one another, or having interactive
discussions. There will not be time for "cover" all material in a
lecture format so you will need to read and learn some topics on your
own from the book, web sources, (or otherwise).
SCHEDULE: The following is a the start of a tentative
schedule, that I will update throughout the semester.
3240 (STILL BEING REVISED) LECTURE AND ASSIGNMENT SCHEDULE 

Section 
Date 
Topics 
HW/Quiz Info 
Prologue & Ch. 1
 8/31T
 Overview of Number Theory
 Read Ch. 2 on Mathematical Induction for Quiz
 §3.12
 9/2R
 Divisibility in Z
 QUIZ on Math Induction

 9/7T
 Common Divisors; Euclid's algorithm


 9/9R
 Bezout's relation and GCD
 Quiz Rewrite due FRIDAY 9/10 3:00PM; HW#1 due MONDAY 9/13 3:00PM.

 9/14T
 Computations in mods


 9/16R
 Units, residues, divisibility tests in Z
 Quiz #2 on divisibility, simple mod computations.

 9/21T
 More divis. tests in Z; Congruences obstructions to
diophantine eqns.


 9/23R
 Factorization and primes in Z
 HW#2 due Fri 9/24 @3:00PM

 9/28T
 Unique Prime Factorization in Z and in R[t]


 9/30R
 Linear systems of congruences
 Quiz #3 on divisibility tests & primes

 10/5T
 Congruences in R[t]; sums of two squares in
Z.


 10/7R
 Powers in mods: Thms of Fermat & Euler
 HW #3 due Fri 9/24 @3:00PM

 10/12T



 10/14R

 Quiz #4

 10/19T
 Orders and repeating decimals
 Attempt Practice Midterm by 10/21R

 10/21R
 Catchup & Review Day
 HW4 due 22 Oct, 3PM

TUESDAY 26 OCTOBER 2010 MIDTERM EXAM 
 10/28R
 RSA Encryption


 11/2T
 Applications of CRT: Phi is multiplicative


 11/4R
 When is 1 a square (mod p)? Gaussian Integers
 HW5 due 5 Nov, 3PM

 11/9T
 Arithmetic in Z[i]


 11/14R
 More Z[i], sums of two squares


 11/16T
 Squares in Z/p


 11/18R
 Counting sums of two squares in Z/p
 Take home Quiz 6 (due 11/30)

2226 NOVEMBER THANKSGIVING BREAK, NO CLASSES 
 11/30T
 Proof of QR via counting "points on spheres"


 12/2R
 Finish proof of QR
 HW6 due 3 Dec, 3PM

 12/7T
 Jacobi symbols & SolovayStrassen Test
 Read Squares
Mod p, III, , § 4

 12/9R
 More applications of Square Patterns
 Read Square Applications
I & II

SUNDAY 12 DECEMBER 4:00PM: REVIEW SESSION IN MSB 215 (Attempt Review Problems by today) 
TUESDAY 14 DECEMBER 1:00PM: FINAL EXAM IN MSB 215

Web Resources
Keith
Conrad has an Expository
paper website with lots of useful handouts, some of which we will
use during the semester. (He also provided the most of the links below.)
The
Prime Pages.
A current list of known
Mersenne primes, ordered by the (prime) exponent. Click
here to join GIMPS
(the Great Internet Mersenne Prime Search).
A discussion of
Euclid's algorithm. There are links to other items from
number theory at the bottom of the page.
Biographies of
Mersenne,
Fermat,
Euler,
Gauss,
Dirichlet,
and
Riemann.
An
interview
with JeanPierre Serre, one of the most prominent
number theorists of the 20th century.
Here's an Online Mind
Reader. Can you figure out how it works?
NEWS & NOTES
Back to my home page.
Here is
a Practice Midterm (typos
corrected 10/19)
Here are
Roby's Rules for Rewrites .
Here's
the wiki for Math
453 at UICC (Fall 2008), where the students filled in an outline
of webpages created by the instructor. Perhaps you'll find the
explanations here useful?
Here is
a Review & Practice Final

