matthias beck

MATH 310 Elementary Number Theory Fall 2024

Lecture	Mon/Wed/Fri 9:00-9:50 a.m. Burk Hall 46
Prerequisites:	MATH 301 with a grade of C or better or consent of the instructor
Instructor	Dr. Matthias Beck
Office	Creative Arts 32
Office hours	Mondays 11-12, Wednesdays 10-11, Fridays 1-2, by appointment, and via zoom

Course objectives. Number Theory studies the integers: numbers like 0, 1, -3, 34, ... which we used since childhood. A fundamental concept is that of divisibility: the integer a divides the integer b if we can find an integer c such that b=ac. This simple concept gives rise to a beautiful theory, encryptions schemes which are used on any computer today, and many famous open problems in mathematics, among other things.

As an example, we will consider prime numbers: those integers >1 that are only divisible by themselves and 1. There are infinitely many prime numbers--one of the many theorems we will prove in this course. The only even prime is 2, all others are odd. To say this in a more sophisticated way: there are infinitely many primes that have a remainder of 1 when divided by 2. Can one say something similar when we divide primes by 3? The only prime that does not give a remainder when divided by 3 is 3 itself. So all others give a remainder of 1 or 2. Are there infinitely many primes in both cases? To give an open problem, we mention twin primes: those are pairs of primes that differ by 2. Are there infinitely many twin primes?

Syllabus. Topics in this course will include:

• Divisibility
• Primes
• Congruences
• Arithmetic functions
• Primitive roots
• Quadratic reciprocity
• Continued fractions

Textbooks.

• George Andrews, Number Theory, Dover, 1994 (reprinted from the 1971 original).
• William Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer, 2008.

Participation. A nontrivial part of the material covered in this class will be worked out in small groups during class sessions. It will thus be essential that every student participates actively in every class. If you have to miss a class due to a medical or family emergency, please let me know before the class; otherwise, I expect you to be in class and actively engaged.

Homework. I will assign homework problems as we go through the material. You may (and should) work together with your class mates. We can discuss the homework problems at any time during class, and you can hand any of your solutions for feedback. We will have a homework quiz every Wednesday at the beginning of class, in which you will be asked one definition and one problem given in the previous week.

SageMath. You will be expected to use the open math software sage in some of your homework assignments. Here is a good introduction to sage. I will add useful sage commands for our class here.

Grading system.

Homework quizzes	50%
Midterm Exam (October 14, in class)	25%
Final Exam (December 16 2:45 p.m. or December 18 12:30 p.m.)	25%

I want to ensure that each of you accomplishes the goals of this course as comfortably and successfully as possible. At any time you feel overwhelmed or lost, please come and talk with me.

The math. The way to learn math is through doing math. It is vital and expected that you attend every class meeting. You will get a good feel for the math from there, but it is even more crucial that you do the homework. Working in groups is not only allowed but strongly recommended. Our class is based on Federico Ardila's Axioms:

1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
4. Every student deserves to be treated with dignity and respect.

Fine print.
SFSU academic calender
BS rule
Academic Integrity and Plagiarism
Tutoring
CR/NCR grading
Incomplete grades
Late and retroactive withdrawals
Students with disabilities
Religious holidays

This syllabus is subject to change. All assignments, as well as other announcements on tests, policies, etc., are given in class. If you miss a class, it is your responsibility to find out what's going on. I will try to keep this course web page as updated as possible, however, the most recent information will always be given in class. Always ask lots of questions in class; my courses are interactive. You are always encouraged to see me in my office.