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I'm a math major and a magician. If you would like to, you could call me a math-magician because I solve math equations magically. :-P
Also, to fulfill my my very first assignment, I would like to...
Andy Schultz is the professor for the course and the admin of this Wiki. Andy grew up in Texas and Mississippi before moving to Davidson, North Carolina for college. Despite his plan to major in...
My name is Anna. I am a junior math major and secondary education minor here at U of I. I am from Chicago. By my picture, it is clear to see that I am a Cubs fan. That is my boyfriend in the...
Each of the following problems is worth 1 point if you answer it completely correctly. Partially correct answers will receive zero credit. These extra point problems must be turned in to me by 11am...
The following problem can be turned in for bonus on the second midterm. The first was originally a homework problem, but a typo in the original assignment might have left lots of people confused....
The first chapter of the book introduces you to the basics of number theory, getting you comfortable with playing around with integers, introducing you to primes, and asking some basic questions...
Chapter 2 is all about modular (or congruence) arithmetic and how it can help us understand numbers and their properties. Here's the stuff we covered in class
Lecture 5: Primes in Arithmetic...
Chapter 3 develops the concept of an arithmetic function (pronounced "air - ith - meh - tick"), which we motivated by studying Euler's function. These functions are some of the most important —...
Chapter 4 is all about determining which numbers modulo p (where p is an odd prime) are actually squares of other numbers modulo p. This culminates in Gauss's amazing Law of Quadratic...
Chapter 5 is all about orders modulo m. The big theorem in this chapter is the theorem of the primitive element, which characterizes those moduli m for which a primitive root exists.
Lecture 21: A...
Chi Wang is a senior in computer science. He was born and raised in Taiwan until high school. He then came to US with his older brother and lived in his uncle's house while studying in Naperville,...
The professor for this course — Andy Schultz — is available to talk to you about anything you'd like, pretty much whenever you'd like. His office is Room 238 of Illini Hall, and his office...
Here we'll collect notes for all the lectures given in class. I'll fill in the skeleton of these notes, but you'll be placed into groups which are responsible for filling out anything which is left...
Introduction
It doesn't hurt to give a brief intro to the topics. It doesn't need to be very complex, but should give the reader a taste of what's to come.
Here's what a basic coursenotes page...
Class Profiles » Dan Bergren
Dan Bergren is currently an undergrad student studying mathematics at the University of Illinois at Urbana-Champaign. He also runs a personal blog, Wandering with...
What fun!
I am from about three hours north of here near the illustrious yet overcrowded Six Flags: Great America. I am quite interested in the teaching of mathematics around the college level and...
Summary
We started class by going over the midterm grades. We then discussed , the function which adds up the divisors of a given integer n. We saw that was multiplicative and we gave a formula...
Hi, my name is Efrain Guzman and I am senior majoring in math. I was born in the south side of Chicago and therefore a Sox fan. My favorite sport is basketball, which I enjoy to watch and play....
Having covered Chapters 1 through 5, we're now have the basics tools necessary to talk about pretty much any "elementary" number theoretic results. Students will be responsible for presenting group...
This is a fake homework assignment, which means that you do not have to complete this assignment nor turn it in. It is provided here so you'll have an opportunity to practice some of the ideas...
Greg Gifford is a junior in Math at the University of Illinois. He grew in the western suburbs of Chicago, in Glen Ellyn. His biggest influence in helping him decide to major in math was his high...
Overview
You and your teammate will be asked to prepare a project on a number theoretic topic of your choosing. The topic should be something which your team finds interesting and which you can...
Note: you should read the full assignment prompt before getting started.
Your 0th assignment is to create your own profile for the wiki; you should have this done by Tuesday night. You can include...
This assignment is due by the beginning of class on Wednesday, September 3rd. Be sure to review the homework guidelines before getting started.
Prove that for any and any positive integer , at...
This assignment is due to my office by 11am on Wednesday, November 5th. Be sure to review the homework guidelines before getting started.
Using only Lagrange's Theorem and Fermat's Little Theorem,...
This assignment is due to my office by 11am on Wednesday, November 19th. Be sure to review the homework guidelines before getting started.
Create your own public key (your primes should be greater...
This assignment is due in class by 11am on Friday, December 5th. Be sure to review the homework guidelines before getting started.
Using Pari or Mathematica if necessary, determine two values of n...
Now that you have completed your project, your final homework assignment is to write a brief evaluation (1 or 2 pages) of the project, giving feedback on the assignment itself, the topic you chose,...
This assignment is due by the beginning of class on Wednesday, September 10th. Be sure to review the homework guidelines before getting started.
Prove that if is prime and , then .
For an integer...
This assignment is due by the beginning of class on Wednesday, September 17th. Be sure to review the homework guidelines before getting started.
From Strayer, Chapter 1, do the following...
This assignment is due by the beginning of class on Wednesday, September 24th. Be sure to review the homework guidelines before getting started.
Prove that for any prime number p, the following...
This assignment is due to my office by 3pm on Tuesday, September 29th. Be sure to review the homework guidelines before getting started.
From Chapter 2 of Strayer, do the following...
This assignment is due to my office by 3pm on Wednesday, October 8th. Be sure to review the homework guidelines before getting started.
From Chapter 3 of Strayer, do the following...
This assignment is due to my office by 11am on Wednesday, October 15th. Be sure to review the homework guidelines before getting started.
Evaluate in the following cases
For a certain...
This assignment is due to my office by 11am on Wednesday, October 22nd. Be sure to review the homework guidelines before getting started.
Suppose that p is an odd prime and that . If we let...
This assignment is due to my office by 11am on Wednesday, October 29th. Be sure to review the homework guidelines before getting started.
From Chapter 4 of Strayer, do the following...
Mathematics is best learned by doing, so part of your responsibility in this class is to complete weekly homework assignments. The basic rules for submitting homework are:
Your assignment should...
For pages which you're allowed to edit, simply click on edit button at the bottom of the page. This will open an editor. Above the textbox you'll see an array of buttons which will help you format...
As a means for creatively engaging with the course material, you'll be expected to make regular posts to the forum section of the course Wiki to record number theoretic ideas or conjectures which...
Jane Kinas was born and raised in the town of Harvard, IL. She's currently a senior in mathematics and ready to graduate, despite not having a very clear picture of what to do afterwards. During...
Hi, my name is Jen Berg and I'm a junior majoring in math and minoring in cs. I grew up on the north side of Chicago and eventually plan to move back there after grad school. I work for the NetMath...
My name's John Lundeberg, but most people can't pronounce that, so just call me Lundy. I'm a junior, math major, in the secondary ed program, and I'm a member of Acacia Fraternity.
I'm from...
JohnMark Lau is a senior in Computer Science. He has lived in Illinois his whole life, and is currently living with his parents in Mahomet and commuting to school every day. When on campus, you can...
Please change this page according to your policy (configure first using Site Manager) and remove this note.
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I am a senior in Mathematics with a minor in Religious Studies. I grew up in the middle of corn and bean fields, also known as Heyworth, IL (you could, until this last year, stand at one of the...
Summary
We started today by getting to know the policies and expectations in the course. All of this is available already on the syllabus, but if you have any questions don't be shy about...
Summary
Today we're finishing up our discussion of Fermat's Little Theorem. By the end of the class we'll state and prove a generalization which works for all integers — not just primes.
A Test...
Summary
In today's class we began by talking about a few "magic tricks" one can do using modular arithmetic. These were motivated by a posting from the forum which showed a video of a...
Summary
Today we talked a little more about the Euler function, then moved on to talk about arithmetic functions more generally. We saw a few examples of arithmetic functions, but we spent the...
Summary
In today's class we started by giving an alternate proof of the fact that . We then discussed the function , which counts the number of positive divisors of n. We saw that the function is...
Recap and Summary
Last class period we introduced the function, which takes a given integer n and adds up the divisors of that number. In today's class, we discussed perfect numbers: those numbers...
Summary
Today we began class by discussing some of the relevant details for the upcoming Group Projects. We then talked a little more about odd perfect numbers. The majority of the class was...
Summary
In today's class we finished talking about convolutions, with the highlight being the proof and a few applications of the Mobius Inversion Formula. Afterwards we talked about quadratic...
Summary
In today's class we began by talking about quadratic residues once again. We noticed that we could list the quadratic residues by squaring the first residues mod p, and that there were an...
Summary
In today's class we began by reviewing the criteria we saw for evaluating Legendre symbols last class period, particularly when the "numerator" was either -1 or 2. Afterwards we set to...
Summary
Today we continued our discussion of divisibility and its basic properties. This culminated in the division algorithm. We also got a preview of the topic for next class: prime...
Summary
Today we talked more about quadratic reciprocity. We started by giving some more applications of this powerful theorem. We then concluded class by giving a proof of Eisenstein's Lemma, a...
Summary
Today we finished our discussion of quadratic reciprocity by providing a proof of the theorem which is based on Eisenstein's Lemma. Afterwards we began talking about the notion of order for...
Summary
Today we continued our discussion of the order of an integer a modulo m. We discussed many arithmetic properties of order, including its relationship to as well as how one can predict the...
Summary
Today we started our search for those integers which do have a primitive root. We began by considering the case of a prime number p. The main tool we used was an analysis of the properties...
Summary
Today we stated and proved the primitive root theorem, giving a full description of those integers m for which there exists a primitive root modulo m. This involved showing that certain...
Summary
We started class by finishing the proof of the Primitive Root theorem: first by reviewing the case when the modulus is , then using this to find primitive roots mod . Afterwards we saw an...
Summary
Today we spent time discussing techniques for checking whether a given number is a primitive root for a given modulus. Hopefully this cleared up some confusion that people had when they...
Summary
Today was our last day discussing Chapter 5, and we spent the bulk of the day discussing the idea of index relatively to a given primitive root mod m. The index is akin to the logarithm...
Summary
Today we discussed the results of the second midterm, including how your raw score correlate to a letter grade. Students who received below a "C" were told how they could receive some...
Summary
Today we took a sneak peak at the kind of number theory which comes after a typical "elementary" course like ours. Much of our time was devoted to studying properties of the famous Riemann...
Review & Summary
Yesterday we spent the class delving deeper into the depths of divisibility, seeing a few more basic examples and then working a couple of more complicated examples. We also...
During this lecture we discussed many classifications of classical numbers. These numbers included perfect, amicable, and sociable numbers. We discussed many different types of perfect numbers,...
Summary
During this lecture we discussed many classifications of classical numbers. These numbers included perfect, amicable, and sociable numbers. We discussed many different types of perfect...
Rational and irrational numbers
To start this talk about , it would be best to extend the integers to the rational and irrational numbers since this is a number theory class.
We denote the set of...
The Fibonacci Numbers
1 1 2 3 5 8 13 21 34 …
Virtually everyone in the world of Mathematics has either heard of or encountered the Fibonacci Numbers in...
Introduction
Fermat's Last Theorem states
(1)
has no integer solutions for any .
This problem had been unsolved for centuries, making it one of the longest standing challenges to mathematicians...
Pythagorean Theorem
The the square of the length of the hypotenuse of a right triangle is equal to the sum of the square of the lengths of the other two sides. If c denotes the hyponuse & a and...
Summary
Today we discussed a variety of topics concerning continued fractions. This included finite and infinite continued fractions, convergents, as well as applications of each.
Finite Continued...
Introduction
We began class by stating that our topic falls under the topic of Diophantine equations.
Definition: A diophantine equation is any equation with one or more variable to be solved in...
Introduction
The Know/Don't know problem is a riddle that uses topics we learned in Chapter 1, namely divisibility and prime numbers. The guidelines:
X and Y are mystery numbers, such that X>1...
Recap and Summary
Last class period we talked about a handful of primality topics. These were the first steps in showing that prime numbers are the building blocks of the multiplicative structure...
Recap & Summary
Last class period we talked about greatest common divisors and least common multiples. We saw some of the basic properties of both, and we also saw that they are intimately...
Summary
In today's class we'll wrap up our discussion on the Fundamental Theorem of Arithmetic, eventually hitting Dirichlet's Theorem on primes in an arithmetic progression. Afterwards we'll...
Recap & Summary
In the last class period, we defined the important notion of congruence modulo an integer m: . We finished by talking about complete residue systems module m, and we showed that...
Recap & Summary
Last class period we spent some time asking the question
When does have a solution?
Today we're going to continue talking about linear congruence equations, in particular...
Recap & Summary
Last class period we talked about solving linear congruence equations. Among other techniques, we discussed multiplicative inverses and how they can be used to help solve...
Summary
Today we started by introducing a few methods for solving simultaneous congruences suggested by students in the class. We then finished our proof of Wilson's Theorem before moving on to...
Abundant and Deficient Numbers
We notice that perfect numbers take the form of , but what about the numbers such that ?
Well, obviously, if , then either or . Actually, there are names given for...
John Lundeberg
Mike Mitchaner
Mallory Moncivaiz
Summary:
This presentation will provide multiple classifications of classical numbers. Such numbers include perfect, amicable, and sociable numbers....
Hi, I'm Mallory. I'm taking Math 453 as part of my math major. I'm in the secondary mathematics education program and hope to someday become a high school math teacher. I'm a member of the service...
I have lived in Urbana, IL my entire life, so naturally why would I want to leave for college? I am currently a senior in Mathematics and hope to become a high school teacher after graduating. I...
Hello class, my name is Brian and I do math. I do math for my major, for my country, and for fun. fun fun fun math is. Just kidding! In all seriousness though I am a senior, planning on teaching...
Phil is a Senior in the College of Liberal Arts and Sciences majoring in Mathematics and minoring in Computer Science. He was born in the city of Chicago, IL, but now resides in Niles, IL, a suburb...
Wikidot uses a markup language called LaTeX to generate properly typeset mathematics. Since you'll be posting to the forum and creating Wiki pages with serious mathematical content, it's good that...
Introduction
Number theory is all about integers, primarily their properties under multiplication and addition. The cornerstone of the multiplicative theory is divisibility, so this first section...
Welcome to Math 453
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Introduction
Number theory is the study of integers, primarily their structure under the operations of multiplication and addition. Though this seems a humble beginning, it is surprising how...
( 16 points) Complete the following sentences
Dirichlet's Theorem states…
For a given integer n, counts…
(15 points) Answer the following questions either “true” or “false.” If...
Give complete, concise answers to the following prompts. Be sure to include all hypotheses.
State either Gauss' Lemma or Eisenstein's Lemma; be sure to indicate which lemma you have chosen to...
There will be two midterms in the course and a final. The midterms are scheduled for Wednesday, October 1st and Friday, November 7th; if these dates change for some reason, you will receive at...
From the Chicago suburb of Buffalo Grove. Currently live in an apartment on Third Street between John and Daniel. Ran on the University of Illinois Cross Country and Track teams for two years,...
Hey, I'm a senior majoring in Mathematics, with a concentration in Mathematics. I like to golf, and long strolls on the beach … just kidding, but seriously, I do like golf. I used to be a Math...
Forum
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Coursenotes
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Extra Material
Course...
If your team is able to come up with a topic on your own, that's great! Send me an email and I'll make sure that you "reserve" your topic. Project topics will be assigned on a "first come, first...
Introduction
This is the course Wiki for the Fall08 manifestation of Math 453, Section D13. This wiki is only available to authorized users, which basically means students registered in the class....
According to Wikipedia, the world largest wiki site:
A Wiki ([ˈwiː.kiː] <wee-kee> or [ˈwɪ.kiː] <wick-ey>) is a type of website that allows users to add, remove, or otherwise edit...
Hi, I am Yuji. I was born in Tampa Florida. I moved to the Chicago suburbs when I was 7. Now, I currently reside in Schaumburg, IL. Mathematics and Music have always been my strongest points. I...
