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Number Theory
Questions or Comments?
- Send me email (click here and delete "QQQ")
- Homepage: http://www.math.uconn.edu/~troby
- Office: MSB M212; Phone: 860-486-4433
- Office hours: Tues/Thurs 1:00-1:50 in MSB 212 and by appointment. Half my time is spent running the Q Center, so I'm frequently elsewhere. I'm happy to answer questions or schedule appointments by email, which I check frequently.
Class Information
COORDINATES: Classes meet Tuesdays and Thursdays 2:00-3:15 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=978-0-470-42413-1.
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 full-fledged progamming environment, check out the free open-source 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:
- Division theorem in Z and R[T]
- Divisibility and Greatest Common Divisors
- Modular Arithmetic
- Decimal Data
- Squares Modulo Primes
- Primes and Congruence Conditions
- Moduli with a Generator for the Units
- Chinese Remainder Theorem
- Original RSA Paper
- Gaussian Integers
- Squares mod p parts I, II, and III
- Square Applications I
- Square Applications II
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 HW/Quiz Info Prologue & Ch. 1 8/31T Overview of Number Theory Read Ch. 2 on Mathematical Induction for Quiz §3.1-2 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) 22-26 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 & Solovay-Strassen 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 Jean-Pierre Serre, one of the most prominent number theorists of the 20-th 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