CRN 13114
From Classes
The best way to learn is to do; the worst way to teach is to talk.
Paul Halmos (1916-2006)
Contents |
Syllabus for Math 5370c (Number Theory and Algebra)
- Time and Place: MW 18:30-19:50 in Bell Hall 130A
- Instructor: Dr. Helmut Knaust, Bell Hall 219, tel. 747-7002, hknaust@utep.edu
- Office Hours: After class, or by appointment
- Textbook: Number Theory Through Inquiry, by David C. Marshall, Edward Odell and Michael Starbird. MAA Textbooks, The Mathematical Association of America 2007.
- Prerequisites: The course requires a certain level of mathematical maturity that you have gained by having thoroughly and successfully grappled with the basics of logic and proofs in MATH 5370a or MATH 3325.
- Course Objectives: The main focus will be on Number Theory. You should expect (and I will expect) that you make considerable progress in the following areas:
- Become familiar with the fundamental results of Number Theory. Highlights of the course include the Fundamental Theorem of Arithmetic, the Chinese Remainder Theorem, Fermat's Little Theorem, and hopefully Public Key Cryptography (RSA);
- Thoroughly understand the definitions of the basic concepts of Number Theory such as divisibility and modular arithmetic;
- Continue to develop your ability to use the method of proof to establish these fundamental results.
- Be able to present and defend a proof to a group of your peers.
- Be able to recognize a rigorous proof when you read or see one. Conversely, be able to pick out the weak spot(s) in a less rigorous argument. Be able to fill in details in a sketchy proof.
- Once you have devised a proof, be able to write it down in a clear, concise manner using correct English and mathematical grammar.
- In-class Activities and Presentations: Mathematics is not a spectator sport. Therefore I will not give lectures. I will call on students to give solutions to the problems in the textbook on the blackboard. Your presentations are the most important part of the course. Your chances of passing the course without spending a significant amount of time on preparing in-class demonstrations are zero: Your in-class work, evaluated for both quality and quantity, will account for 40% of your grade.
- Tests: A midterm exam will be given on October 15. This exam counts 30% of your grade.
- Final Examination: The final exam on December 10 at 19:00-21:45, is comprehensive and mandatory. It counts 30% of your grade.
- Time Requirement: I expect that you spend an absolute minimum of six hours per week outside of class on reading the textbook, preparing for the next class, reviewing your own notes, and completing assignments. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.
- Attendance: Due to the course structure, attendance is mandatory. An unexcused absence will result in a problem grade of 0 for the day of the absence. Five absences (excused or unexcused) will lead to dismissal from the class with a grade of "F".
- Students with Disabilities. If you have a disability and need special accommodation, please contact the Disabled Student Services Office (DSSO) in Union East 106, 747-5148, dss@utep.edu.
- Academic Integrity.: All students must abide by UTEP's academic integrity policies, see http://academics.utep.edu/Default.aspx?tabid=23785 for details.
Ground Rules
- The textbook contains numerous problems. Students will solve these problems at home. The instructor will then call on students during class to present solutions to these problems on the blackboard.
- When in the audience, students are expected to be actively engaged in the presentation. This means checking to see if every step of the presentation is clear and convincing, and speaking up when it is not. When there are gaps in the reasoning, the students in class will work together to fill the gaps.
- The instructor serves as a moderator. His major contribution in class will consist of asking guiding and probing questions. He will also occasionally give short presentations to put topics into a wider algebraic context, or to briefly talk about additional algebraic concepts not dealt with in the textbook.
- Students may use only the textbook and their own notes taken during the semester; they are not allowed to consult other books or materials. Students must not talk about assignments to anyone other than class participants and the instructor. Students are encouraged to collaborate with other class participants; if they do, they must acknowledge other students’ contributions during their presentation. Exemptions from these restrictions require prior approval by the instructor.
- The instructor is an important resource. He expects frequent visits from all students in class during his office hours – many more visits than in a “normal" class. Among other things, students probably will want to come to the instructor’s office to ask questions about concepts and assigned problems, they will probably occasionally want to show the instructor their work before presenting it in class, and they probably will have times when they just want to talk about the frustrations they may experience.
- It is of paramount importance that we all agree to create a class atmosphere that is supportive and non-threatening to all participants. Disparaging remarks will be tolerated neither from students nor from the instructor.
Class Progress
- 12/1/08: 3.13
- 11/26/08: 3.11, 3.17, 3.20, 3.22
- 11/24/08: 3.16, 3.17, 3.18, 3.19
- 11/19/08: 3.14, 3.15. Groups report out next time.
- 11/17/08: 2.18, 3.8, 3.9
- Write up 2.38 for next Monday!!
- 11/12/08: 2.38, 3.5
- 11/10/08: 2.48, 3.1, 3.2, 3.3, 3.4
- 11/5/08: 2.28,2.42,2.43,2.44
- 11/3/08: 2.41
- 10/29/08: 2.32, 2.33, 2.34, 2.35, 2.37
- 10/27/08: 2.9, 2.12, 2.28(1/2)
- 10/22/08: 2.26, 2.27, 2.29, 2.30, 2.31
- 10/20/08: 2.22, 2.23
- 10/15/08: Midterm Exam
- 10/13/08: 2.8, 2.13, 2.20
- 10/9/08: 2.3 \(\Leftarrow\), 2.14, 2.15, 2.19, 2.21
- 10/6/08: 2.3 \(\Rightarrow\), 2.7, 2.10, 2.11; turn in 2.7 next time.
- 10/1/08: 1.38, 1.40, 1.48, 2.1
- 9/29/08: 1.39, 1.42, 1.43, 1.45
- 9/24/08: 1.33, 1.36, 1.37, 1.41
- 9/22/08: 1.26, 1.29-1.32
- 9/15/08: 1.23, 1.25, 1.27, 1.28
- 9/10/08: 1.18, 1.21, 1.24
- 9/8/08: 1.17-1.22
- 9/3/08: 1.11-1.16
- 8/27/08: 1.2-1.10
- 8/25/08: 1.1
Open Problems
- 2.46
Materials
- Mathematica notebooks: The function Π(n) and the Prime Number Theorem | The Sieve of Eratosthenes | The Euclidean Algorithm
- Visualization of the Sieve of Eratosthenes for numbers up to \(601^2\).
- Talk at the 2008 GEPCTM Conference
- It’s Easy to Get Interested in Number Theory. An Interview With Karl Rubin (MAA Focus 28 (6))
- GIMPS=Great Internet Mersenne Prime Search
- A list of all known Mersenne Primes
- A New Formula for Generating Primes. Ivars Peterson, The Mathematical Tourist, August 8, 2008.