matthias beck

professor of mathematics
kumi does math
san francisco state university

MATH 310

Elementary Number Theory

Fall 2022

cartoon: foxtrot does number theory

Lecture Mon/Wed/Fri 2:00-2:50 p.m. TH 211
Prerequisites: MATH 301 with a grade of C or better or consent of the instructor
Instructor Dr. Matthias Beck
Office Thornton Hall 933
Office hours Mondays 4-5, Wednesdays 11-12, Fridays 10-11 & by appointment

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:

Textbooks. I will neither require nor closely follow a specific book. However, it is extremely useful to read alongside our class; two books I recommend are

Participation. Most of the material covered in this class will be worked out in small groups during class sessions; this philosophy is sometimes called inquiry-based learning. 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 certain problems as homework problems; they will be due before following Wednesday class. You may hand them in early for feedback. You are expected to work together with your class mates, but each of you will submit their own solutions. If you typeset your problems, you may hand them in via email. I recommend LaTeX for mathematical typesetting; here is a guide if you've never used LaTeX and here are some tricks that I've learned over the years.

Sage. You will be expected to use the open math software sage in some of your homework assignments. Here is a good introduction to sage.

Grading system.

Participation 10%
Homework 50%
Midterm Exam (October 17) 20%
Optional Midterm II (November 16)
Final Exam (December 12, 2:45 p.m.) 20%

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
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.

department of mathematics
san francisco state university
1600 holloway ave
san francisco, ca 94132