Math 4181: Elementary number theory

Information about this course

• Text book: Number Theory: An Introduction to Algebra, by Fred Richman.
  The course book is available as a course pack from the COOP bookstore, 3960 Burbank Drive.
• Class times and place: MWF 2:40 - 3:30am, Locket 135
• Office hours: MWF: Locket 210. 3:30 - 4:30am, or by appointment.
• Contact info: Office phone: 578 1603. Email: Verrill@math.lsu.edu
• Homework: Homework will be given each week. This will be ungraded. It is important to do the homework, since test and exam questions will be based on homework questions. Homework will regularly be discussed in class.
• Tests: (in class time)
  • test 1: Friday October 5
  • test 2: Friday November 16
• Final Exam: Saturday December 15, 7:30 - 9:30 am
• Computation of final grade:
  • 25% test 1 (covers chapters I, II)
  • 25% test 2 (covers chapters III, IV)
  • 50% final exam (covers chapters I, II, III, IV, V)
• Grading scale: 90-100%=A, 80-90%=B, 70-80%=C, 60-70%=D.
• Make up policy: No tests or exams can be dropped, but make up exams are possible in the case of illness or unavoidable absences, e.g., for job interview, or clash with exams for other courses, provided you have documentation to prove your reason for absence. This should be given in advance of the day of the exam where possible.
• Course web page: http://www.math.lsu.edu/~verrill/teaching/math4181/index.html

• Course overview:

  Number theory starts off with the study of whole numbers, the most basic objects in mathematics. So number theory is one of the oldest branches of mathematics. Ancient Greeks, Egyptian, and Babylonians studied basic problems about numbers, such as whether there are infinitely many prime numbers, or how to find integer solutions of equations like x² + y²=5z². See Saint Andrews University page on Pythagoras's theorem in Babylonian mathematics for pictures of some of the oldest known number theory.

  Because whole numbers are discrete, i.e, not continuous, e.g., there is no whole number between 0 and 1, number theory is very relevant for computers, which work digitally, i.e., also with discrete quantities, e.g., 0 and 1.

  Because of this, Number theory is the basis of important computer applications, such as cryptography, used to transfer data securely over the internet, and coding theory, used to transmit data, e.g., from your CD to your CD player, in such a way that errors in transmission can be corrected.

  So, as well as being an old area of mathematics, number theory is a very important and active area of modern mathematical research.

  Number theory is a very broad area. This course will concentrate on algebraic aspects of number theory. There will be some overlap with some topics covered in the abstract algebra course, MATH 4200, but nothing about abstract algebra will be assumed, though if you are taking that course, you may find this course gives useful extra examples of topics covered in MATH4200.

• Course contents:

  In this course, we start with studying the integers, and then move on to studing numbers modulo n, denoted Z_n. These form the basis of RSA cryptography, and also have many other applications in mathematics. The study of Z_n will take up most of the course (this is chapters II, III, IV of the text book). We should then have time to study one further topic, sums of squares, chapter V.

  Approximate schedule:

  • Chapter 1: Integers. (first two or three weeks)
  • Chapters II, III, IV: Numbers modulo n. (most of course)
  • Chapter V: Sums of two squares. (last two or three weeks)
  • Last week of class: Review material covered in course.

• Course study guide:

  We will closely follow the text book, in the order it's written.

  Please try and read and read ahead for each class. This should mean reading on average about two or three pages before each class (there are about 40 classes, and we cover about 80 pages of the text book). Please try and read every word, and make a note of what you don't understand, so you can ask about it in class.
  Expect to cover between 1 and 2 sections per week, depending on the lengths of the sections.
  A strong student who is intending to study number theory at graduate level should aim to work through all the excersise in each section, though probably we will not have enough time to go over every exercise in class.
  There are many many web sites on number theory, reading around will give you different perspectives, so if you have time this is also a good idea.

  This is a pure mathematics course, and proofs will be given, and students will be expected to be able to learn to read and understand proofs and reproduce proofs in tests and exams.
  If there is time, application to RSA cryptography will be explained, but this will not be examined.

Handouts, notes, etc

• Week 1, chapter 0:
  • My notes for class on August 29, 2007 (second class). This class was based on going through problems at the end of chapter 0. In the class, we went through pages 1, 2, 3, and problem 8 on pages 6 and 7. These notes include solutions to the other problems at the end of the chapter. The notes give a brief introduction to proof and logic. The notes probably go into more detail than we really need right now, but hopefully answer questions people might have about some of the exercises.
  • Notes for class on August 31, 2007 (third class). These correspond to section 1.1 of the text book. These notes include solutions to exercises, which we'll go over next Wednesday. Make sure to try to do the exercises yourself before looking at the notes, and come to class with any questions you have.
• Week 2, chapter 1, sections 1.1, 1.2
  • Class for September 5, 2007 (fourth class) (week 2): Today we went over some of the exercises (see previous lecture notes), and over page 10 and half of page 11 of the text book. Next class we should be able to finish section 1.1, and look at some of the exercises at the end of 1.1, so please try some of these in advance.
  • Notes for classes on September 5 and 7, 2007 (second week of class). Also includes solutions to exercises, to be covered in class on Monday September 10. Try as many as you can on your own first.
• Week 3, chapter 1, sections 1.2, 1.3
  • Notes for class on September 10, 2007 (third week of class). See notes for last class for the solutions to exercises, also covered in this class. Try reading through as much of 1.3 and the exercises for 1.3 for next class.
  • Note for classes on September 12 and 14, 2007. Also includes solutions to exercises from section 1.3
  • Try this "practice test" at home to test if you've mastered the material in chapter 1.
• Week 4, chapter 2, sections 2.1, 2.2
  • Note for classes on September 17, 2007.
    We covered half of section 2.1. To give an example of a result we will later prove, using congruences, we discussed Fermat's little theorem (which is on page 35 of the text book), and Fermat Pseudoprimes. To give an idea of how effective primality testing based on the result of Fermat's little theorem is, we discussed how many primes there are up to some limit, and how many Fermat Pseudoprimes there are. If you're interested in these topics, there are many websites about primes, e.g., Chris Caldwell's The primes pages, especially the "how many primes are there?" page, and about pseudoprimes, e.g. Richard Pinch's page on pseudoprimes. See Mathworld pages on Fermat pseudoprimes for basic definitions of Fermat pseudoprimes.
    Please to read to the end of section 2.1, and try as many exercises from the end of section 2.1 for next class.
  • Note for classes on September 19, 2007. Includes solutions to exercises at the end of section 2.1.
  • Section 2.2. Note for classes on September 21, 2007. If you've not seen modular arithmetic before, you may find it useful to construct addition and multiplication tables modulo n.
    For examples, see Cut the knot modular arithmetic page. On this page you can make addition and multiplication tables modulo N for N from 2 to 30. (contains a java applet, so may be slow to load.) There is a list of interesting patterns to observe in the tables.
• Week 5, chapter 2, sections 2.2, 2.3
  • Note for classes on September 24, 2007.
  • Note for classes on September 26, 2007.
  • Note for classes on September 28, 2007.
  • Practice for test, given out on Friday: Practice test 2.
• Week 6, chapter 2, sections 2.3, 2.4, and test on Friday
  • Monday October 1: We did exercises from section 2.3. See notes for class last Wednesday and Friday.
    Also went over practice test solutions. See Practice 1 solutions and Practice 2 solutions
  • Wedensday: Note for classes on October 3, 2007.
  • Friday: First test, covering chapter 1 and 2 up to and including section 2.3.
• Week 7: Section 2.4
  • Monday: Went over test solutions, and part of section 2.4, notes here.
  • Wednesday notes, including solutions to exercises in 2.4.
• Week 8: section 3:1
  • Monday October 15 notes, on RSA public key cryptography.
  • Wednesday October 17 notes RSA example, and section 3.1.
  • Friday October 19 notes, section 3.1, exercises.
  • Hand out from Friday: Tables for multiplication in U_n for n from 2 to 15. Which of these are isomorphic to each other? We'll go over the answer on Monday.
• Week 9: Section 3.2
  • Monday October 22 notes, on section 3.2, direct sums of abelian groups.
  • Wednesday: Wednesday October 24 notes, material to end of section 3.2, with some hints for exercises.
  • Friday: Friday October 26 notes, notes on subgroups of Z_N,+. (we did some parts of this again on Monday.)
• Week 10
  • Monday October 29 notes, on when is Z_N,+ a non trivial direct sum, and solutions to exercises for 3.2.
  • Wednesday October 31 notes, section 3.3.
  • Friday November 2 notes, section 3.4.
    List of what to expect on the second test, probably the second test will also include the Chinese remainder theorem, given in Chapter 4, section 1.
• Week 11
  • Monday November 5 notes, end of chapter 3 (see Friday's note's) and start chapter 4.
  • Wednesday November 7 notes, 4.1 and some of 4.2.
  • Friday November 9 notes, 4.2.
• Week 12,
  • Monday November 12, 2007, we spent the class time going over solutions to practice test questions. See Practice test 3 solutions.
  • Wednesday November 14 notes, started 4.3.
  • second test, Friday November 16. Test and solututions here.
• Week 13
  • Monday: Notes on 4.3
  • Wednesday: End of 4.3, also solutions to exercises for 4.3, done in class .
  • Friday: Holiday for thanksgiving
• Week 14
  • Covered some 4.4. Notes for Monday November 26.
  • Expect to cover some of chapter 5: Notes for Wednesday November 28.
    Handout for final exam preparation
  • Parts of chapter 5, outline of when an integer is the sum of two squares. Notes for Friday November 30.
• Week 15
  • This week is a review. We will go through the Solutions to Handout for final exam preparation.
    (This handout is not yet complete; will be updated shortly.)
    On Monday we covered questions 1 to 8.
• Week 16: Exam week. Final exam, Saturday December 15.