Math 404/504, Spring 2022
Meeting times: W,F from 10:30–11:45am
Textbook: Silverman, A friendly introduction to number theory
Homework: View homework problems
Office hours: Thursday at 2pm and by appointment
Number theory is the study of patterns in the set of natural numbers and its arithmetic operations. Like geometry, it is one of the most classical fields in mathematics. In number theory there are many questions which simultaneously have very simple statements and extremely complicated solutions (or no known answer at all). In this course we will aim to explore a variety of different areas of inquiry and problem-solving methods in number theory.
Anticipated learning outcomes
- Possess and exhibit a working knowledge of number theory content
- Read and write proofs in number theory
- Explore mathematical questions, experiment using data, formulate conjectures
- Appreciate abstract mathematics, number theory, and its applications
- Weeks 1-3: Divisibility and congruences
- Weeks 4-6: Primes and primality testing
- Weeks 7-9: Quadratic reciprocity and sums of squares
- Weeks 10-12: Pell’s equation and Diophantine approximation
- Weeks 13-15: Sequences, continued fractions
The course will be delivered through synchronous remote class sessions. Each session will include a mixture of questions on previous material, discussion of new material, and in-class problem solving.
In order to pass the class with a C, you are expected to attend class sessions and make a reasonable attempt at all weekly homework questions by the due date.
Attendance Please attend remote class sessions, be prepared to ask questions and discuss the material, and collaborate with your peers on in-class activities. You may miss a few sessions without penalty, but additional absences must be documented.
Homework Assignments will be given each week and collected the following week. You must attempt all problems. Any problems which are not sufficiently correct or complete must be resubmitted later using your portfolio (see below).
To earn a grade of B or A, you must maintain a portfolio of supplemental homework problems. These problems will be posted regularly, and may also be suggested by the student in consultation with me. If you are enrolled in Math 404, try to collect 5-10 supplemental problems for a B and 10-15 supplemental problems for an A. If you are enrolled in Math 504, try to collect 10-15 supplemental problems for a B and 15-20 supplemental problems for an A. (These numbers are for the entire semester, not each portfolio evaluation!) Your portfolio will also include required resubmissions from homework assignments.
Portfolio evaluation dates:
- February 11
- March 18
- May 2
What is cheating?
You are encouraged to collaborate with your peers, and you are welcome to use online resources when you are stuck (please reference). But please keep in mind that you must always fully understand your solutions and most importantly write them in your own words.
This syllabus is subject to change. I may make refinements and updates during the first few weeks of the semester. While I don’t expect any substantial changes, please allow for some flexibility. I will give notice before making any changes to the syllabus.