Math 5020

Fundamentals of Mathematics for Teachers

Description: Number Theory and Combinatorics are branches of mathematics in which theorems and problems are usually easy to state but often difficult to prove or resolve. This course will deal with topics in these two fundamental mathematical fields, including modular arithmetic, linear and quadratic diophantine equations, continued fractions, permutations and combinations, distributions and partitions, recurrence relations, generating functions, formal power series.  The use of computers for mathematical exploration will be encouraged through the computer program Maple.  The course will cover material from 'Number Theory' by George Andrews and supplementary material on generating functions and species.

Professor Mike Zabrocki
Office: TEL 3046
Hours: Thursday 4-5:30pm
phone: 416-736-2100 x33980
e-mail: zabrocki at mathstat dot yorku dot ca
Class meets Thursday 6-9pm in VC 102
Text : Number Theory by George Andrews
A random picture by M. C. Escher that looks cool

We have a FORUM for math 5020.  This is a web site that will allow you to post questions/comments/communications about the course.  I will try to keep track of postings and answer them on a regular basis.

Forum problems for Fall:

Course presentation topics
Fermat's Little Theorem
Kevin Smith
Sept 29
Wilson's Theorem
Shirley Ting
Oct 6
card shuffling
Pierre Lacoste
Oct 20
solving congruences
Dorota Mazur
Oct 27
chinese remainder theorem
Pauline Fu
Nov 3
multiplicative functions
Mike Eden
Nov 10
Möbius inversion
Samia Saleh Nov 17
primitive roots
Jeff Irwin
Nov 24
distrib. primes & Tchebychev's ineq
Toni Katsinos & Carol Miron
Dec 1
quadratic reciprocity
Melissa Giardina & Paul Attar
Jan 5
pseudo-primality testing Jacobi and Legendre symbols
Ada Tsui & Andrea Young
Jan 12
RSA and Digital Signatures
Taravat Moshtagh
Jan 19
Sum of 4 squares theorem
Keith Auyeung
Jan 26
Generating functions
Anisoara Preda
Feb 2
Jacobi triple product identity
Anna Yoon
Feb 9
Fermat's Last Theorem and A. Weil
Margarita Panayotova
Feb 16
Ferrer's diagrams and partition ident.

partitions which fit in a rectangle

Grades for the course will be based on the following criteria:

Class presentation
Un-exams (4 x)
Forum exercises Fall 15%+Winter 15%

We've been following the book very closely.  I'll try to keep track of what sections we've covered.

Sept 12 - Sect 1.1, 1.2, 2.1
Sept 19 - Sect 2.2, 2.3
Sept 26 - Sect 3.1, 3.2
Oct 6 - Sect 3.3, 3.5, 4.1, 4.2
Oct 20 - Sect 4.3
Oct 27 - Sect 5.1, 5.2
Nov 3 - Sect 5.3
Nov 10 - Sect 6.1,6.2,6.3
Nov 17 - Sect 6.4


March 30, 2006- Now we have the tools, we can write down the generating functions for every set of partitions under the sun (worksheet #9). You should be able to do all of these exercises once you know what a Durfee square is. It is the largest square which fits inside the diagram for the partition. Every partition with a Durfee square of size k x k can be decomposed (uniquely) into a tuple consisting of a Durfee square, a partition of length <= k and a partition with parts of size <= k. (sorry, but the following picture is upside down compared to what we have been doing in class...also height = length).
Most of them take 30 seconds to write down the answer. Just glancing through it, number 19 is one of the few challenging ones on this list.

March 23, 2006- I announced the last unexam. This will be due for next week. If you cannot get it to me by then you will have to send it to me by email (NO Microsoft Word documents with equations please) no later than April 7.

March 23, 2006- We will consider partitions as combinatorial objects and dissect these to write down generating functions (worksheet #8).

March 9, 2006- I have been handing out roughly one set of problems and this week was no exception. We showed that some combinatorial identities (worksheet #7) could be proved by taking coefficients in generating function expressions. We did these in class.

March 2, 2006- I gave you a handout with some combinatorial problems which can be solved by generating functions. Worksheet #6. We will do more of these, but make sure that you understand (10), (11), (13), (14), (15), (16) which we did in class. For homework I asked you to do (10) and (11) on WS #4 because we didn't get to it that night.

For homework I asked people to prove (10) and (11) on Worksheet #4
I also handed out a worksheet consisting of a few expressions that I want to take coefficients (Worksheet #5) in. Please do these exercises for next week.

February 16, 2006- I had very few people do the exercise that I assigned at the end of class last week and so I assigned it as the next unexam problem.
Unexam #3 is due February 23, 2006.  Sorry for the short notice but we are coming close to the end of this course and I was expecting this to be an easy question that everyone had an answer for this week.

February 16, 2006- We are slowly building up connections between three classes of mathematical concepts
three classes of math concepts
Making these connections will require building up certain skills.  This next set of exercises makes the connection between class 1 and class 3.  Each one of these should take you no more than a couple of minutes.  Please take the time to do all 26 of them for next week.
Worksheet 3: Algebraic expressions and sequences
The other worksheet I gave out was a list of a library of generating functions that you can refer to.  We will prove all of these.  So far Anasoria proved (1) and (9) and I proved (2) in class
Worksheet 4: Examples of generating functions

February 2, 2006-
I gave a matching worksheet which we did half of in class.  The purpose of this exercise is to make connections between algebraic expressions in  variables a0, a1, a2, a3,.... and combinatorial descriptions of objects.  I use word 'widget' to represent 'some combinatorial object.' An example of a 'widget of size n' might be 'a partition of the integer n into parts of size at most 3' or a 'a word of length n in the letters a and b such that a's are separated by at least two bs.'
Worksheet 2: Sequences and sets of objects

January 26, 2006-
I gave worksheets about the online-encyclopedia of integer sequences.  To everyone who was there I gave them a question off of this worksheet.  If you did not get one, please ask me for a question.  We will come back to this when we know more generating functions.
Worksheet 1: Sequences and sets of objects

January 20, 2006
- Here is the Microsoft word document that Taravat gave copies of last night (the text version of the powerpoint presentation). I will try to convert it to PDF later.

January 18, 2006- For tomorrow nights class we will be going to the Gauss lab because Taravat will be doing a presentation using Maple. Please be in the classroom ON TIME because we will leave from there and go to the computer lab.

January 6, 2006- Last night I announced the next 'unexam.' The first of these unexams was an exercise that was quite straightforward and I didn't see a lot of variation on the assignment. This one I should see much more.

November 17, 2005- Last week I announced the first 'unexam.' I wanted to write some expectations (as well as the question) and place it on this web page. This assignment is due November 24, 2005.

I've had to make some changes to the schedule of presentations this term. As you might know Inna dropped the class suddently and so we don't have someone for November 17. If a brave soul would like to take up her topic or date then I wouldn't have to feel uncomfortable about not having time for everyone next term. If not, we will make due.

Dorota has been practicing using excell to compute greatest common divisors, exponentiation and chinese remainder theorems using Excel. I am posting her worksheets in Chinese Rem theorem PDF exponentiation PDF and Chinese Rem theorem Excel and exponentiation Excel format here. I like the use computers to do where possible because they are efficient ways of doing mathematics and in order to explain to a computer how to compute something, you must understand that computation yourself on some level.

I don't remember who asked to do the 3 of a kind, straight and flush in class. I'll ask you next week to fill them in. Get them done though because I had one solution already posted but not by the people who asked for them.

Oct 25 - The web server holding this web page crashed and some information on this page may have been lost. Please let me know if you notice anything missing.

I've decided not to emphasize the use of computers although I may do occasional demonstrations in class (as I did on section 3.5).  The reason for this is that it takes enormous amounts of time to learn a computer language.  I would like the people who do the presentation topics to consider using the computer when they think it is possible.  I am willing to help in any way I can (and we can also reserve a computer lab if you want to take the whole class to do experments on the computers).  If you want to learn a computer program you should take a look at the program MuPad.