|
|
| Quick links to: | Announcements | General information | Assignments | Notes | References | External resources |
10/26/06 Midterm review problems posted I've posted review problems for the second midterm in the Notes section below. I've also posted the second midterm exams from the previous two years. Solutions to the review problems will be posted next week.
10/20/06 Online calculator The most recent homework includes a problem (6.2.15) that requires computing large powers. There's an online calculator on the GMP library homepage that you may find useful for this.
10/11/06 ROOM CHANGE! From Monday October 16th onwards, the 4pm section of Math 1020 will meet in Thackeray 525 instead of Thackeray 703. (The room for the 10am section is unchanged.)
10/09/06 Pollard Rho slides The slides used to present the Pollard Rho method, with minor additions and corrections, are in the notes section below.
09/29/06 Midterm I solutions posted I've posted solutions to the first midterm.
09/22/06 Midterm I reminder A reminder that the first midterm will be held during class time on Wednesday September 27th. I've posted solutions to the review problems below, as well as solutions to the first midterm exams from the previous two years.
09/15/06 Midterm review problems posted I've posted a list of review problems for the first midterm in the Notes section below. I've also posted the first midterm exams from the previous two years. Solutions will be posted next week.
09/13/06 New Mersenne prime discovered! The 44th known Mersenne prime was discovered by participants in the GIMPS project on September 4, 2006. More details at the GIMPS home page.
09/06/06 Calculus Graders needed From Dr. Beatrous: We will need to hire some undergraduates to help with homework grading in calculus and precalculus courses. Interested undergraduates can obtain an application from Carol in Thackeray 301.
08/30/06 Assignment 1 The problems for Assignment 1 are listed in the Assignments section below; the assignment is due at the beginning of class on Wednesday, September 6.
08/28/06 Assignment 0 due September 1 Assignment 0 is due by 5pm on September 1.
08/28/06 Course web page now online! The course web page now exists! This web page will be your central resource (outside of classes) for information about the course; please come here often to look for assignments, announcements and other information.
Course description Elementary number theory is the study of properties of the integers that can be proved using only elementary techniques; that is, without making use of other branches of mathematics such as analysis or geometry. In this course, we'll study the notions of divisibility, primes, greatest common divisors, congruences, primitive roots, quadratic residues and quadratic reciprocity. We'll also study a variety of present-day applications of number theory, including various aspects of cryptology.
Prerequisites Math 0430: Introduction to abstract algebraic systems (groups, rings, etc.) is the main prerequisite. Students will also be expected to have the skills necessary to solve problems and write rigorous proofs. No previous knowledge of number theory is required.
Assessment The final grade will be composed of grades for coursework, two midterm exams, and a final exam, in the following proportions.
Coursework will consist of a series of weekly homework assignments. There
will be between 10 and 12 of these in total. Homework will be assigned every
Wednesday, and completed assignments will be
collected at the beginning of class on the following Wednesday. You're
encouraged to discuss homework problems with each other, but you should write
up your solutions individually. Any use of internet resources should be
clearly acknowledged. Homeworks should be presented neatly and
carefully.
Late homeworks will not be accepted without prior
arrangement: if for some reason you think you won't be able to get your
homework in on time, please come and discuss this with me before the
due date.
Exam dates According to the registrar, the final exam for section 1040 (MWF at 10:00) will be on Thursday December 14 at 12:00pm, while the final exam for section 1020 (MW at 4:00) will be on Wednesday December 13 at 2:00pm. There will be two midterm exams, provisionally scheduled for Wednesday September 27 and Wednesday November 8. The midterm exams will be held during class time. Please mark these dates in your diaries!
Textbook The textbook for the course is `Elementary Number Theory and Its Applications' by Kenneth H. Rosen, 5th edition. ISBN 0-321-23707-2. More information and supplementary materials are available on the publisher's website, at http://www.aw-bc.com/rosen.
Syllabus Here's a rough syllabus for the course. Please bear in mind that the information below isn't exact: some sections listed may be only partially covered, and we may cover additional material not listed below. When exam time comes around your course notes should serve as the primary reference. Therefore it's especially important to make sure that you obtain, read and understand the class notes in the event that you have to miss a class.
| Week beginning | Textbook sections | Topics to be covered |
|---|---|---|
| 08/28 | 1.1–1.3 | The integers; well-ordering and the principle of induction |
| 09/04 (no class Monday) |
1.3–1.4 | Fibonacci sequence and recursive definitions; complete induction |
| 09/11 | 1.5, 3.1–3.2 | Divisibility; primes |
| 09/18 | 3.3–3.5 | The Euclidean algorithm and the fundamental theorem of arithmetic. |
| 09/25 | 3.7, 4.1–4.3 | Congruences and the Chinese Remainder Theorem; Midterm I |
| 10/02 | 4.6, 5.1, 5.5 | Applications of congruences |
| 10/09 | 6.1–6.3, 7.1 | Fermat's Little Theorem; Pseudoprimes; Euler's theorem; the Euler phi function |
| 10/16 | 8.1, 8.4 | Cryptology |
| 10/23 | 9.1–9.3 | Orders; primitive roots |
| 10/30 | 9.4–9.5 | Index arithmetic; primality tests |
| 11/06 | Ch 10 (selected) | Applications of primitive roots; Midterm II |
| 11/13 | 11.1–11.3 | Quadratic reciprocity |
| 11/20 (no class Wednesday or Friday) |
11.4, 11.5 | Applications of quadratic reciprocity |
| 11/27 | Further applications | |
| 12/04 | Applications; Review for final exam |
Assignments will be posted in this section. All assignments are due at the beginning of class on the indicated due date.
| Assignment | Problems | Due date | |
|---|---|---|---|
| Assignment 0 | See detailed description. | Friday September 1, by 5:00pm | |
| Assignment 1 | Section 1.1: 5 (this one isn't easy!) Section 1.2: 3, 24 Section 1.3: 2, 7, 8, 11, 20, 24, 30 |
Wednesday September 6 | |
| Assignment 2 | Section 1.3: 14 (see example 1.24) Section 1.4: 4, 7, 8, 14, 16, 39 |
Wednesday September 13 | |
| Assignment 3 | Section 1.5: 5, 11, 23, 26, 34, 37 Section 3.1: 2, 6, 10, 13 Section 3.2: 4, 10, 13 |
Wednesday September 20 | |
| Assignment 4 | Section 3.3: 2, 8, 22 Section 3.4: 4 Section 3.5: 2, 4, 66 |
Wednesday September 27 | |
| Assignment 5 | Section 3.3: 14 Section 3.4: 19, 20 Section 3.5: 14, 16 (first part only) Section 3.7: 2, 7, 8, 16 Section 4.1: 8, 9, 20 |
Wednesday October 4 | Selected solutions |
| Assignment 6 | Section 4.2: 2, 3, 9, 15, 18 Section 4.3: 4, 7, 12, 22 Section 4.6: 1, 2 |
Wednesday October 11 | |
| Assignment 7 | Section 5.1: 3, 4, 17 Section 5.5: 8, 12, 13 Section 6.1: 12, 21, 40, 41, 43 |
Wednesday October 18 | |
| Assignment 8 | Section 6.2: 2, 8, 15, 18 Section 6.3: 4, 6, 10, 15, 20 Section 7.1: 2, 3, 6, 13 |
Wednesday October 25 | |
| Assignment 9 | Section 8.1: 2, 6, 12 Section 8.4: 1, 7 Section 9.1: 3, 4, 9, 13 Section 9.2: 6, 9, 16 Section 9.3: 2 |
Wednesday November 1 | |
| Assignment 10 | Review problems for the second midterm! | No homework due | |
| Assignment 11 | Section 9.4: 1, 2, 5, 8, 9, 10 Section 9.5: 2 Section 13.3: 1, 2 |
Wednesday November 15 | |
| Assignment 12 | Section 11.1: 2, 4, 6, 10, 11, 14, 25 | Wednesday November 29 |
I'll put digital copies of any handouts or overheads used during the course in this section, along with other course-related materials. Most of these are PDF files; you may need to download a copy of Acrobat Reader in order to view them. Please do let me know if you have any problems viewing these files.
Here's a list of some of other books related to the class material, that you might find interesting or useful.
| [D] | H. Davenport. The Higher Arithmetic (7th edition). Cambridge University Press, 1999. ISBN 0-521-63446-6. A very readable introduction to elementary number theory and beyond! |
| [MOV] | Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone. Handbook of Applied Cryptography. CRC Press, 2001 (5th printing). ISBN 0-8493-8523-7. An excellent reference for many of the cryptographic protocols discussed. |
| [M] | Richard A. Mollin. Fundamental Number Theory with Applications. CRC Press, 1998. ISBN 0-8493-3987-1. Covers much of the same material as the course text; worth a look if you want another viewpoint. |
| [TW] | W. Trappe and L. Washington. Introduction to Cryptography with Coding Theory. Prentice Hall, 2002. ISBN 0130618144. |
Here are some external links of varying relevance to the course. Let me know of anything that you'd like to see added here! A reminder: treat all information coming from any of these web pages with some caution; there's a lot of good information there, but errors and inaccuracies abound.
Return to Mark Dickinson's home page