University of Connecticut College of Liberal Arts and Sciences
Department of Mathematics
Tom Roby's Math 3240 Home Page (Fall 2012)
Number Theory

Questions or Comments?


Class Information

COORDINATES: Classes meet Tuesdays and Thursdays 2:00‒3:15 in MSB 215. The registrar calls this Section 001.

PREREQUISITES: Math 2710 (Transition to Higher Math)

TEXT: Instead of a text, we'll use course handouts.

WEB RESOURCES: The homepage for this course is http://www.math.uconn.edu/~troby/math3240f12/.

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.

GRADER: The grader for this course is Ricky Martin, Office MSB 230, Office Hrs: MW 8:30‒9:30AM, Tues/Thurs 9:30‒10:30.

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 THURSDAY 18 OCTOBER 2012 in MSB 215. 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 2--3 students, with only one set of solutions handed in per group. I've gathered (tentative) of HW Policies here, currently a list of seven. Please hand in HW to the envelope outside MSB 230.

Many of the assignments will reference Handouts available here, though I've also put some direct links in below:

Here are the assignments:

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. 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 handouts, 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
Date Topics HW & Quiz Info
8/28T Overview of Number Theory Read handout on Mathematical Induction for Quiz
8/30R Division theorem in Z and Divisibility Quiz #1 on Math Induction
9/4T Common Divisors, GCD, & Euclid's algorithm
9/6R Bezout's relation and GCD HW #1 due MONDAY 9/10 5:00PM @MSB-230 (UPDATED!)
9/11T Computations in mods
9/13R Units, residues, divisibility tests in Z Quiz #2 on divisibility, simple mod computations.
9/18T More divis. tests in Z; Congruences obstructions to diophantine eqns. READ two posts by Tim Gowers on Why isn't FTA obvious? and Proving FTA
9/20R Factorization and primes in Z Q1Rewrite due today in class; HW #2 due Fri 9/21 @2:00PM
9/25T Unique Prime Factorization in Z and in R[t]
9/27R Linear systems of congruences Quiz #3 on divisibility tests & primes
10/2T Congruences in R[t]; sums of two squares in Z.
10/4R Powers in mods: Thms of Fermat & Euler
10/9T Compositeness Tests & Carmichael Numbers HW #3 due Fri 10/12 @2:00PM in MSB 230
10/11R Logarithms & Orders
10/16T Catchup & Review Day Attempt Practice Midterm by today
THURSDAY 18 OCTOBER 2012 MIDTERM EXAM
10/23T Orders and repeating decimals Read carefully from Order Handout: 2.1, 3.1-5, 3.11-15, 4.5-7
10/25R More on Orders
11/2R Class cancelled by SuperStormSandy Quiz #4 on orders in groups of units due. (postponed until 11/4R).
11/4R RSA Encryption (original paper.) Quiz #4 on orders in groups of units due; HW #4 due 5 Nov, 5PM
11/6T When is -1 a square (mod p)? Gaussian Integers
11/8R Arithmetic in Z[i] Midterm Rewrites due in class today.
11/13T More Z[i], sums of two squares in Z HW #5 due Fri 16 Nov @2:00PM in MSB 230;
11/15R Squares in Z/p Read Squares Mod p, I,
19-23 NOVEMBER THANKSGIVING BREAK, NO CLASSES
11/27T Counting sums of two squares in Z/p Read Squares Mod p, II,
11/29R Proof of QR via counting "points on spheres" Read Squares Mod p, III, , § 4
12/4T Jacobi symbols & Solovay-Strassen Test HW #6 due 5 Dec, 3PM
12/6R More applications of Square Patterns Read Square Applications I & II
SUNDAY 9 DECEMBER, 3:00: REVIEW SESSION IN MSB 215 (Attempt Review Problems by today.)
TUESDAY 11 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 many 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


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