MAT 341 Number Theory

Welcome to the home page of Prof. Xiao Xiao’s Number Theory course at Utica University. You can find all the informtion and documents for this course on this page. Please check this page frequently for announcements and assignments.

Important Dates

• Add/Drop deadline: 1/19/24
• Spring break: 3/11/24 - 3/15/24
• Withdraw deadline: 3/25/24
• SOOT: 4/22/24 - 4/29/24
• Final exam: 8:00am, 5/2/24

Instructor Information

• Instructor: Prof. Xiao Xiao
• Email: [email protected]
• Office: White Hall 255
• Office hour: Tuesdays and Thursdays 1-2pm, Wednesdays 10-11am or by appointment.

An Important Course Policy

I pride myself on having a good environment for working and learning. It is very important to me that we all treat each other with care and respect, in equal measure. I know that I ask students to take risks in class almost every day, and this can be challenging for many. I ask that you help me keep our classroom a supportive place for each of the people in it. Each of us deserves the space to bring our full, authentic selves to class and be comfortable. (Adapated from T.J. Hitchman.)

General Course Information and Policies

• Course name: MAT 341 Number Theory

• Course credit hours: 3-credit

• Course prerequisite: MAT 305, or permission of instructor

• Class time and location: Tuesdays and Thursdays 8:30-9:45am, Wednesday 11:30-12:20 (will be used for special projects or make up classes) at Hubbard Hall 205.

• Textbook: We will use Number Theory Through Inquiry, by David C. Marshall, Edward Odell, and Michael Starbird. The ISBN of the book is 978-1-4704-6159-1.

• Course description: Elementary number theory including prime numbers, greatest common divisors, congruences, modular arithmetic, Fermat’s and Euler’s Theorem, and its related application in RSA cryptography.

• Program learning goals: In accordance to the learning goals of the Department of Mathematics of Utica University, MAT 401 will introduce and reinforce students’ ability of:
  • (PLG1) Reading and analyzing mathematical proofs.
  • (PLG2) Writing mathematical proofs.
  • (PLG5) Communicating mathematics in written form.
• Course learning objectives: Upon successful completion of this course, students will be able to:
  • understand and prove basic results involve greatest common divisors;
  • understand and apply fundamental theorem of arithmetic;
  • understand modular arithmetic and be able to solve linear congruences equations;
  • understand and apply Fermat’s little theorem, Euler’s theorem, and Wilson’s theorem;
  • understand RSA public key cryptography.

• Class organization: This course will likely be different from any other math course you have taken before. As an instructor, I will not be lecturing most of the time although I love lecturing very much. Scientific research shows that most people do not learn mathematics by listening, instead, they learn by doing it! I am sure you have said to yourself before “It looked so easy when the professor was doing it, but now I am confused when I have to do it by myself.” Why? Because the knowledge belongs to your professor and does not belong to you. You do not learn the knowledge simply by hearing it once or twice from somebody else. In order for you to have a more thorough understanding of the knowledge, we will incorporate ideas from an educational philosophy called the Moore method (after R. L. Moore). More precisely, we will use the modified Moore method, also known as inquiry-based learning. Most of the time during the class, students will be presenting proofs of theorems that they have produced by themselves, and not by other people or textbooks. A significant portion of your grade will be determined by how much mathematics you produce.

• You should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course. On the other hand, you may use each other, the course notes, me, and your own intuition.

• Regular attendance is mandatory and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your participation grade.

Daily Homework and Presentations

• After each lesson, I will assign a list of the problems that we are working from the textbook. You will be expected to read and solve (or make as much progress as possible) these problems before walking into the next class period. This will ensure that you are ready to take an active part in our class presentations and discussions. You do not need to submit your daily homework but you are welcome to email me your draft for feedback before the class.

• As a team, each student will be responsible presenting 1/n of the total problems where n is the total number of students in the class. Students will be informed before the beginning of the semester the list of problems that they will be responsible for.

• Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken very seriously since they spearhead the work of the class.

• In order to make presentations go smoothly, presenters need to write out the proof in detail and go over the major ideas and transitions, so that they can make the proof clear to others.

• The purpose of presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the proof or the solution clear to the other students.

• Presenters need to write in complete sentences, using proper English and mathematical grammar.

• Fellow students are allowed to ask questions at any point and it is the responsibility of the presenter to answer those questions to the best of their ability.

• Since the presentation is directed at the students, the presenter should frequently make eye contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

• Daily homework is meant to prepare students for the presentations during the next class. Students will be evaluated based on their presentations although they do not receive direct credit from doing the daily homework.

Portfolio

• The object is to maintain a current account of the work we do. Every task that we encounter in the class needs to be typed up using LaTeX and is to be included in a (class-shared) portfolio. Each entry in the portfolio is intended to be complete and polished. Do not include scratch work.

• At the end of each week, you should type up the problem that you have presented during the week in the shared portfolio. This is considered the first draft and naturally it won’t be perfect. Therefore there should be further revisions by the rest of the class. Please use LaTeX to type the portfolio. Please use the following Overleaf template as a start.

• Each student will receive a class grade of the portfolio (based on the overall quality of the entire portfolio) and a personal grade of the portfolio (based on the problems from that particular student).

Evaluation

Your final grade will be determined by presentations/participation and portfolio.

Academic Integrity and Collaboration

Collaboration and cooperation are extremely helpful in the learning process and encouraged in most circumstances! However you may only collaborate with students currently enrolled in the same section of the course. When collaboration has occurred, you must acknowledge this clearly (state the name(s) of the person(s) you collaborated with on each problem.) You are permitted to collaborate on big ideas and hints with classmates, however, you must work independently when writing up solutions. All collaborations should occur when your collaborator is at essentially the same stage of the problem solution as yourself. In particular, if you have not yet started a problem and you ask a friend who has already completed it, how do you do this problem. This counts as plagiarism. The resulting work is not and cannot be considered your own. Regarding to outside resources: all work (including daily, weekly etc.), unless directly stated otherwise, the only resources you may use are our course textbook and your class notes. You are not permitted to go looking for completed solutions to problems in other texts or resources. In particular, use of internet resources is completely off limits for homework problems. If you see a solution, there is no way that you can claim to have an original solution. Evidence of using internet sources in your work will result in a minimum penalty of failing the assignment. Copying a solution, or any part of a solution, from any source (friend, internet, book, etc.) in any setting, constitutes plagiarism. You may not seek the help of an instructor or tutor (other than me) unless you first discuss this with me in advance. If you do not verify that this is acceptable before seeking help, this will be considered plagiarism as well. I am always willing to discuss any aspect of the course with you. Evidence of dishonest behavior on any assignment will be grounds for a minimum penalty of failing the entire assignment. In severe cases, the minimum penalty will be failure of the course. Peers who willingly assist others in acts of plagiarism are equally guilty, and will suffer similar penalties. All academic dishonesty will be reported to the Academic Standards Committee. There might be additional sanctions by the Academic Standards Committee such as dismissal from the university. See Utica University official page for Academic Honesty for more details.

Special Accommodation

The stuff just below is the University approved language, and is a bit… “legalese:. The point is, if you need accommodations to succeed in this course, talk to me and we can make sure you get what you need. And the social environment of this course is important to me, too. Let’s work together to make a welcoming and affirming space for everyone.

Any student who has need of special accommodations in this class due to a documented disability should speak with me as soon as possible, preferably within the first two weeks of class. You should also contact Judy Borner, Director of Learning Services in the Academic Support Services Center (315-792-3032 or [email protected] ) in order to determine eligibility for services and to receive an accommodation letter. We will work with you to help you in your efforts to master the course content in an effective and appropriate way. See Utica University official page for Office of Learning Services.

Disclaimer

It is the students’ responsibility to keep informed of all announcements, syllabus adjustments, or policy changes during the semester via this web page or via school emails. The author of this syllabus reserves the right to change it with notice at any time during the semester.