## Math 115A - Number Theory (Section 2) Fall 2018

Instructor: Brian Osserman

Lectures: MWF 3:10-4:00, Hoagland 168.

CRN: 28639

Office: MSB 3218, e-mail:

Office Hours: M 2-3, W 4-5

Textbook: Kenneth Rosen, Elementary Number Theory and Its Applications (6th edition)
Note: This textbook is very expensive, and is currently in short supply. You may find that the digital version is more affordable.

TA: Cameron Bjorklund

Discussion: R 4:10-5:00, Olson 106

TA Office Hours: R 12-1, MSB 3202

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.

### Exams

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.

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.