Grading: 30% homework, 25% midterm, 45% final exam

Homework: Assigned weekly, due each Friday in class

Welcome to Math 115A: Number Theory

Number theory is a very classical branch of mathematics, going back as
far as ancient Greece and China. For most of its history, it was viewed
as the sort of mathematics one does purely for fun, with no interest in
or prospect of applications, but since the development of
number-theory-based codes in the 1970's, it has become a crucial component
of modern cryptography. Part of the appeal of number theory is its
elementary nature; while a certain amount of mathematical maturity is
necessary for this course, it will not directly use anything beyond
basic high school algebra, and lacks the abstraction of abstract (or even
linear) algebra.

Lecture notes

I will post notes here to supplement the textbook as seems appropriate.

Axioms for the integers:
This gives a list of axioms for the integers, and some proofs of very
basic properties.

Diophantine equations:
A somewhat briefer version of the material in lecture on Diophantine
equations.

Z/mZ as
a number system: This note introduces the idea of thinking of
congruence classes modulo m as an alternative number system.

The Miller factorization
algorithm: This note describes the Miller factorization algorithm
and its relation to RSA, and gives a sketch of why it works efficiently.

Lecture schedule

For those who are following in the book, here is a summary of which
book sections were involved in which lectures. However, I typically do not
cover everything in a given section, and there may be some material not
included in the book.

9/26: syllabus, introduction, and axioms for integers (see
Appendix A of book, and above handout).

9/28: Parts of section 1.3.

10/1: Parts of sections 1.3 and 1.5.

10/3: Parts of sections 1.5 and 3.1.

10/5: Parts of sections 1.5 and 3.3.

10/8: Most of section 3.4, bit of 3.7.

10/10: Sections 3.4, 3.5.

10/12: Bit more of section 3.5, then 3.1.

10/15: Section 3.7.

10/17: More section 3.7.

10/19: Section 4.1.

10/22: More of 4.1.

10/24: Section 4.2.

10/26: Section 4.3.

10/29: Examples and remarks based on Section 4.3.

10/31: The integers modulo m as a number system.

11/2: Midterm exam.

11/5: 4.4.

11/7: 4.4.

11/9: 6.1.

11/26: 6.3 and 7.1.

11/28: 7.1 and 8.4.

11/30: End of 4.1, 6.2.

12/3: More 6.2, Miller factorization.

12/5: Parts of 3.6, 4.6 and 6.1.

12/7: Review.

Problem sets

Problem sets will be posted here each Friday, due the following Friday
in class. You are encouraged to collaborate with other students, as
long as you make sure you understand your answers and they are in your own
words. You are not, under any circumstances, allowed to get answers to
problems from any outside sources.

Only a selection of problems will be fully graded from each problem set.
To minimize
resulting randomness of scores, your lowest problem set score will be
dropped when calculating your grade.

There will be one in-class midterm exam, on Friday, November 2. It
will cover all material up through and including the lecture on Friday,
October 26, corresponding to the first five homeworks, and through
Section 4.3 of the book.

The final exam is scheduled for Monday, December 10, 8:00-10:00 AM in
our usual lecture room.
It will cover all material from lecture, including all 8 homework
assignments, and the additional material on primality testing and
factorization methods covered on 11/30, 12/3 and 12/5.

The median on the midterm was 68%, and the mean was 63%. The letter grade
ranges corresponding to the raw scores are as follows:
75-100%: A range
55-75%: B range
40-55%: C range
30-40%: D range
0-30%: F

Students with Disabilities

Any student with a documented disability (e.g. physical, learning,
psychiatric, vision, hearing, etc.) 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.

Academic Dishonesty

Although I do not expect it to be a problem, cheating (including on
homework) has no place in this class and will punished severely. You can
find the new
Student Code of Academic
Conduct here.

Academic Participation Requirements

Due to new requirements from the US Department of Education, you must
verify your participation
in all your registered classes after instruction begins and before the add
date, which is October 11. Failure to do so can impact your financial aid.

STEM Cafe

The STEM Cafe is organized
by the Women's Resources and Research Center, and offers a place to meet
and discuss math, and has tutoring available even for upper-division classes.