Course content and prerequisites
This course is an introduction to number theory, the part of mathematics that deals with properties of integers and one of the oldest branches of mathematics. The official prerequisite is "Math 241 or equivalent", though more important than being proficient with triple integrals is a sufficient level of "mathematical maturity", as acquired, for example, in proof-based courses such as Math 213, CS 173, or Math 347; in particular, you should know how to properly write up a mathematical argument, and be familiar with standard proof techniques, such as induction and proof by contradiction. This course is a rigorous mathematical course, so there will be a certain amount of definitions, theorems, and proofs. However, there will also be a good deal of concrete, hands-on computations, and many interesting and fun applications.
This course will be significantly different from calculus courses. The solutions to the problems will require more creativity and effort, and in particular, it will not be possible to succeed in this course merely by memorizing the techniques you see in class or in the textbook. To be able to solve the problems in this course, it is hightly recommended that you participate actively in lectures, and put in a lot of effort into solving the homework problems on your own before discussing with others.
The first four chapters in the Strayer text form the core material. We will cover these chapters, and selected topics from the remainder of the text, depending on the time available and student interest.
Elementary Number Theory
, by James Strayer. Written homework assignments and your
independent reading will come primarily from this book.
Weekly written homework
The written homework is a chance to work on problems independently. To succeed in the course, it is important to put in effort into the homework. In particular, before consulting others about a proof problem, you should know the definitions of all the words in the question, have consulted the relevant sections in your class notes and textbook, and spent a significant amount of time thinking about how different concepts in the problem link together.
Be sure that your final write-up is clean and clear and effectively communicates your reasoning to the grader. In particular, proofs should be written in full sentences and contain all the necessary details.
You can discuss the homework problems with others, but you must write up solutions independently, on your own. This is important practice, and will help you do well in the exams in this course. No late assigments will be accepted; however, your lowest two homework scores will be dropped.
The goal of the weekly reading assignments is to help you learn and retain the material. The presentation of the material in the class will usually differ slightly from the presentation of the material in the textbook.
This offering of Math 453 has two 50 minute midterm exams and one 3 hour final exam. The midterms will take place in class. Information on exact locations and times, as well as sample exams, will be posted in the Announcements section.
There will be no make-up exams. Instead, if you miss an exam and have a valid excuse, the exam will be marked as excused. An "excused" exam means that it will not not be taken into account in the computation of your grade. Valid excuses include illness, an out-of-town job interview, etc., and must be documented by a letter from the Dean or Emergency Dean; see the Emergency Dean's website
for more information on this.
Your grade will be computed by combining scores from your written homework, midterm exam, and final exam according to the
|Homework: || 20%
|Midterms: || 20% each, 40% total
|Final Exam: || 40%
You are welcome to bring any questions you have to office hours.
My email is xiannan "at" illinois.edu. I will generally respond to emails within 24 hours.
Homework assignments and rough outlines of homework solutions will be posted here.
Be sure to consult the syllabus for homework policies, especially with regards
to collaboration, write-up standards and submission. The warm-up questions are not to turn in - they are usually quick computational problems with answers at the back of the book. However, they are valuable practice, and you should know how to do these for the exam.
Turn in homework at the beginning of Wednesday's lecture; if you need to miss that class for any reason, please ask someone to bring it in for you or contact me to make alternate arrangements.
Warm-up questions (not to turn in): Section 1.1: 3 a - c, 4, 5, 7, Section 1.2: 17, 21
Questions to turn in: Section 1.1: 2, 10, 11, Section 1.2: 22, 23, 30. Solution Outline
Warm-up questions (not to turn in): Section 1.3: 32 a - d, 34 a-d, 35, 38, Section 1.4: 54 a-d.
Questions to turn in: Section 1.2: 26, Section 1.3: 33 a - b, 36, 41, 42, Section 1.4: 55. Solution Outline
Warm-up questions (not to turn in): Section 1.5: 59 a - d, 60 a-c, 63 a-b, 64, 67 Section 2.1: 1 a-d.
Questions to turn in: Section 1.3: 50, Section 1.4: 56, Section 1.5: 61a-b, 68, 72a-b (the phrasing in the text is slightly misleading; there is a case where there are infinitely many possibilities - for this case, you should state what the possiblities are and justify with proof), 76 (you can use 14a without proof). Solution Outline
Warm-up questions (not to turn in): Section 2.1: 4 a - d, 5, 8, 9, Section 2.2: 28 a-b, 29 a-b.
Questions to turn in: Section 2.1: 3, 13, 17, 20, Section 2.2: 28 e-f, 29 e-f. Solution Outline
Assignment 5 (due Feb 22) :
Warm-up questions (not to turn in): Section 2.3: 33, Section 2.4: 42 a-d, 43, Section 2.5: 50, 51.
Questions to turn in: Section 2.2: 32, Section 2.3: 34, 36, Section 2.4: 44, 46 (assume p>2), Section 2.5: 52. Solution Outline
Assignment 6 (due Feb 29, shorter due to midterm) :
Warm-up questions (not to turn in): Section 2.5: 53, 54, Section 2.6: 66.
Questions to turn in: Section 2.5: 57b-c, 58, Section 2.6: 67. Solution Outline
Assignment 7 (due March 7th) :
Warm-up questions (not to turn in): Section 2.6: 68a-b, Section 3.1: 1, 2, 5a-d, Section 3.2: 10.
Questions to turn in: Section 2.6: 68c-d, 69, 73, Section 3.1: 5g-h, Section 3.2: 9, 14. Solution Outline
Assignment 8 (due March 14th) :
Warm-up questions (not to turn in): Section 3.3: 29, 30, Section 3.4: 41, 42, 43.
Questions to turn in: Section 3.1: 7, Section 3.2: 20, 24, Section 3.3: 35, 39, Section 3.4: 46. Solution Outline
Assignment 9 (due March 28th) :
Warm-up questions (not to turn in): Section 3.5: 52, 53, Section 3.6: 62, 63.
Questions to turn in: Section 3.4: 47, 49, Section 3.5: 55, 57, Section 3.6: 64, 65 (assume p is prime) Solution Outline
Assignment 10 (due April 4th) :
Warm-up questions (not to turn in): Section 4.1: 1, 2, 3, Section 4.2: 12, 13.
Questions to turn in: Section 4.1: 6, 8 (you might want to prove #47 in Chapter 2 - note that we already proved 47a in class so you don't need to prove it again), Sec 4.2: 14d-f, 15, 16, 17. Solution Outline
Assignment 11 (due Friday,April 13th) :
Warm-up questions (not to turn in): Sec 4.3: 28, 29.
Questions to turn in: Sec 4.3: 30, 32, 33. Solution Outline
Assignment 12 (due Friday,April 20th) :
Warm-up questions (not to turn in): Sec 6.3: 13.
Questions to turn in: Sec 4.3: 35, Sec 6.3: 14, 15, 16cd, Sec 6.4: 20, 21. Solution Outline
Assignment 13 (due at the beginning of lecture, Friday, April 27th OR Monday, April 30th, whichever you prefer) :
Warm-up questions (not to turn in): Sec 6.1: 1, 2a-d, 6.2 11a-b, 7.1: 1, 2.
Questions to turn in: Sec 6.1: 3, 6, Sec 6.2: 11c-e, 12, Sec 7.1: 5b-c. Solution Outline
You should read the following sections in the book:
- All of Chapter 1.
- Chapter 2, 2.1-2.4.
- Chapter 2, 2.5-2.7.
- Chapter 3.
- Chapter 4, 4.1-4.2.
- Chapter 4, 4.3, as it is covered in class.
- Chapter 6, 6.1-6.4.
- Chapter 7, 7.1.
Important announcements will be posted here as they come up in class.
D13 & D14: Morning section - first midterm will be held in class on Friday, Feb 24th.
E13 & E14: Afternoon section - first midterm will be held in class on Monday, Feb 27th.
Here is a Sample Midterm.
The solutions to Midterm 1 for the morning section will be described in class on Monday, Feb 27th, 2012. The average was 75%.
The solutions to Midterm 1 for the afternoon section will be described in class on Wednesday, Feb 29th, 2012. The average was 80%.
D13 & D14: Morning section - second midterm will be held in class on Wednesday, April 11th.
E13 & E14: Afternoon section - second midterm will be held in class on Monday, April 9th.
Here is a Sample Midterm.
This sample midterm is slightly more challenging than your real midterm.
We will discuss Midterm 2 in the afternoon section on Wednesday, April 11th. The average was 76%.
We will discuss Midterm 2 in the morning section on Friday, April 13th. The average was 80%.
Our final exam has been decided by Registrar's. The location will be in our normal classroom. The times are below.
Morning section - Friday, May 11th, 1:30pm-4:30pm.
Afternoon section - Friday, May 11th, 7:00pm-10:00pm.
For your convenience, here are your midterms:
Morning section Midterm 1.
Afternoon section Midterm 1.
Morning section Midterm 2.
Afternoon section Midterm 2.
Some of your final questions will come from the midterms and sample midterms posted above as well as the extra sample final questions.
Extra office hours:
May 8th, 2012: 3-4pm.
May 9th, 2012: 2-4pm.