matthias beck

MATH 310 Elementary Number Theory Fall 2025

Lecture	Mondays & Wednesdays 12:30-1:45 p.m. Thornton Hall 404
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, 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.

I will keep a brief diary here which sections we covered each week:

• Week 1: Andrews Sections 2.1-3

The math. The way to learn math is through doing math. One doesn't become a good piano player by listening to good music, and one doesn't learn how to shoot free throws by watching basketball games. Similarly, you don't learn mathematics by watching someone else do mathematics. You will be doing a lot of the math in this course. 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.

Axioms & first principles. 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.

I would emphasize within the last axiom that dignity and respect is also something you should treat yourself to, and that every person in our course, students and professor, are expected to be honest. It is likely that some or all of us will experience challenges and disruptions this semester. Please choose to act with compassion for others and for yourself as we navigate any situations that arise.

Homework. I will assign homework problems as we go through the material. You may (and should) work together with your class mates. You may not search the internet for solutions to problems; we will use our creativity, course texts, and peer collaboration as our tools. We can discuss the homework problems at any time during class, and you can hand in (or bring to my office hours) any of your solutions for feedback. The homework problems from any given week will be due before the following Wednesday class. You can either bring your homework to class or send me a pdf copy via email.

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 have collected some useful sage commands for our class here.

Grading system. In our course, we will use a specifications-based assessment and grading system, which has the following key features:

• There are four elements to assessment and grading: participation, homework, and exams with definition and problem sections.
• There are no points and no partial credit; all assignments will be either be given a pass or an incomplete. This grade, for each problem, will be based on a combination of (a) correctness of the solution and (b) quality of communication in your writing.
• Your class grade will be determined by the amount of course material you have passed, with the requirements to achieve each letter grade specified below. By focusing on learning and mastery instead of accruing points, we can keep our attention focused on what we know and where we need to improve.
• You can revise some of your incomplete work. My goal is for each of you to develop as mathematical thinkers, learners, and collaborators. To support this goal, you are allowed to revise and resubmit selected work during the semester.

The four elements of assessment and grading are as follows.

• Participation: I expect you to be present and engaged every class session. Every student begins the semester with a participation grade of 20 and will lose one point for each unexcused absence.
• Homework: On non-exam Wednesdays, you will submit five homework problems, assigned the previous week. You will be allowed to revise and resubmit them (once) by the following Wednesday.
• Exams: we will have four cumulative exams, on
  • 17 September
  • 8 October
  • 29 October
  • 19 November
  • 17 December
  Each exam will consist of three linear algebra definitions and three problems similar to the homework. You will be allowed to revise and resubmit selected problems (once) within a week of when I return your work.

To earn a particular grade in the class, you need to complete all of the requirements in the column corresponding to that grade in the table below; i.e., if the entry lists x then you need to receive a pass grade on x of those items to earn the corresponding column grade.

	A	B	C	D
Participation (out of 20)	17	15	13	11
Homework (out of 50)	45	41	33	25
Definition exams (out of 15)	13	12	10	8
Problem exams (out of 15)	12	11	9	7

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.

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.