Course Syllabus

Syllabus - Math 115A

Instructor: Eric Severson

Please just call me Eric. My preferred pronouns are he, him his.

Contact: eseverson@ucdavis.edu

Class Time: 2:10-3:50 pm, MTWF on Zoom (see 'Zoom' page on Canvas)

Office Hours: 11am-12 Mon, 6-7pm Wed, 8:30-9:30am Thur

Also I will try to be on every zoom call by 2pm, and can stay on the call as long as people want to ask me questions, so please ask questions then too

Website: All class information will be found on our page on Canvas. The most important tab will be 'Discussions', where I will make a page for every class, where I will post my class notes, the zoom recording, and any other relevant material. Please also use those discussion pages for asking any questions, so that my answers can be helpful to the whole class.

Text: Elementary Number Theory, 6th edition, by Rosen.

One way to access the text is through the 'Bookshelf' tab here on Canvas, but any way you can find access to the book is fine.

Prerequisites: Officially 21B, but we will not actually use very much Calculus. A more helpful prerequisite would have been Math 108 (our Intro to Abstract Math course), since having some experience with math notation and how to write proofs would be helpful.

Course Material: Our study of elementary number theory will focus on understanding the simplest set of numbers, the set of positive integers . Number theory is one of the oldest subjects in mathematics, and is definitely the oldest subject which is still very active for modern research mathematicians. It turns out there is a lot of complicated structure hidden in the prime numbers, and number theory is full questions that are easy to state but hard to prove, or still unproven!

The first part of the course will introduce the basics about prime numbers (Chaper 1 and 3 in the text): divisibility, the Euclidean Algorithm for finding the greatest common denominator, the Fundamental Theorem of Arithmetic that every number has a unique factorization into prime numbers.

The second part of the course will talk about congruences, also known as modular arithmetic (Chapter 4 and 6 in the text), where we treat numbers the same as long as the have the same remainder when divided by $$$$. We will see there is very different algebraic structure here depending on if the modulus is prime or not, and this actually gives some efficient ways to test if large numbers are prime.

The final part of the course will look at applications. One of the most important modern applications of number theory is in cryptography. We will have studied multiple examples of functions that are fast to compute, but hard to invert. You can find two large primes and multiply them together quickly, but trying to invert this and factor their product turns out to be very hard. This is the key behind the RSA algorithm. You can also quickly find very large powers of a number mod p, such as , but trying to invert this (the discrete logarithm) is much harder. We will learn some of the math that let's us do things like exchange money over the internet.

Grading: Your course grade will be based on

15% Classwork: Every day in class, I will ask you to think about at least one problem and turn in your scratchwork before the next class. These will be uploaded on Canvas. This will only be checked to show that you thought about it and wrote something down, you will not be graded for correctness. This is to help incentive you to keep up with the material, which is really important in the fast paced summer session, and get some practice trying to solve lower. If you miss a lecture, you need to watch the video and turn in something for the classwork before the next class. There will be one coursework per day of class, and the lowest 3 will be dropped.

50% Homework: One assignment every week, due at the end of the day every Thursday. The lowest homework score will be dropped.

15% Midterm: We will have a midterm, proctored during class on Friday July 10, so please make sure you are available and can show up to the call early. I know this is not ideal, but from discussion with the rest of the math faculty, this seems to be the only good way to ensure academic integrity. Also, if I can trust that people are not using any other sources than I can comfortably make the exams easier than the homework and test that you have learned the basics of the class.

20% Final: On the last day, Friday July 31, also proctored in class.

Extra Credit Project: There is a lot of cool material in this topic that we are not going to have enough time to go into. Because I want to encourage you to be curious and learn more about any topics that you find interesting, I am going to over an extra credit opportunity where you can write a couple pages or give a short presentation to me about anything related to the class. As we go through the course, I will mention and share material about many topics that you could learn more about for this extra credit. This is also intentionally open ended, so if there is something you want to do for this extra credit, talk to me.

Academic Integrity: There is zero tolerance for cheating in this class, which includes plagiarism. For the homework, you are allowed to discuss the problems with each other if you want, but you have to write the solutions by yourself. You are not allowed to look for solutions on the internet. I have access to the exact same google that you do, and I can and will check for what material exists online to make sure student's solutions did not come from outside sources.

Any cases of academic dishonesty will be taken to the Student Judicial Affairs. Students are expected to have read, to have understood, and to act in accordance with the Code of Academic Conduct, available at http://sja.ucdavis.edu/cac.html.

Summer Classes: Summer courses at Davis are very short. We need to cover what is normally done in 10 weeks plus finals week in only 6. The class will move fast so it is important to stay on top of the material. I highly suggest you don't try to take multiple classes over the summer, it is much more than taking multiple classes during a regular quarter, plus the online format makes it easier to fall behind.

Registration Deadlines: Last Day to Add Classes: Jun 26

Last Day to Drop Classes with Refund: Jun 29

Disability Accommodations: Any student with a documented disability who needs to arrange reasonable accommodations must contact the Student Disability Center (SDC). Faculty are authorized to provide only the accommodations requested by the SDC. If you have any questions, please contact the SDC at 530/752-3184 or sdc@ucdavis.edu.

Mental Health: The unprecedented circumstances we are living in during the pandemic are really challenging for a lot of people, myself included. Students at UC Davis are fortunate to have access to free counseling (https://shcs.ucdavis.edu/online-visits), which I highly recommend you take advantage of. Also, if you are having any difficulties that are interfering with this course, please don't hesitate to reach out.

All future information will be posted on Canvas. It is your responsibility to keep up to date with the material on Canvas. With the exception of test dates and grading policy, any information in this syllabus is subject to change. Important announcements will be sent to your emails through Canvas, so please make sure the email registered there is correct and active.