## Math 115A - Number Theory Fall 2014

Instructor: Brian Osserman

Lectures: MWF 12:10-1:00, Art 204.

CRN: 49316

Office: MSB 3218, e-mail:

Office Hours: M 3-4, F 2-3

Textbook: Kenneth Rosen, Elementary Number Theory and Its Applications (6th edition)
Note: This textbook is very expensive, but used copies are available at much more reasonable prices.

TA: John Murray

Discussion: T 6:10-7:00, Cruess 107

TA Office Hours: T 1-2, MSB 3137

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

Homework: Assigned weekly, due each Wednesday 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.

### 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.

• 10/3: introductory, no specific material.
• 10/6: Appendix A (see also axioms handout above), parts of sections 1.3 and 1.5.
• 10/8: Parts of sections 1.5, 3.1 and 3.5.
• 10/10: Parts of sections 3.3 and 3.4.
• 10/13: Parts of sections 3.4 and 3.5.
• 10/15: Parts of sections 3.5 and 3.1.
• 10/17: Parts of sections 3.1, 3.6 and 3.7.
• 10/20: Rest of section 3.7.
• 10/22: Examples relating to section 3.7, beginning of 4.1.
• 10/24: More of 4.1.
• 10/27: Rest of 4.1, most of 4.2.
• 10/29: Rest of 4.2, some of 4.3.
• 10/31: Rest of 4.3.
• 11/3: The integers modulo m as a number system.
• 11/5: Exam.
• 11/7: 4.4.
• 11/10: More on 4.4.
• 11/12: 6.1.
• 11/14: 6.3.
• 11/17: 7.1.
• 11/19: 8.4.
• 11/21: More on 8.4.
• 11/24: End of 4.1, first half of 6.2.
• 11/26: Rest of 6.2.
• 12/1: 4.6.
• 12/3: End of 6.1, Miller factorization.
• 12/5: 3.6 (Fermat factorization), Dixon's algorithm.
• 12/8: Example of Dixon's algorithm, the quadratic sieve.
• 12/10: More on the quadratic sieve, further developments.
• 12/12: Review.

### Problem sets

Problem sets will be posted here each Wednesday, due the following Wednesday 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.

Some points will be awarded for the total number of completed problems, but 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.

• Problem set #1, due 10/15: do the following exercises from the book: 6 of Appendix A; 20 and 24 of 1.3; 14, 16 and 26 of 1.5; 10 and 14 of 3.3; and 2 of 3.4.
• Problem set #2, due 10/22.
• Problem set #3, due 10/29.
• Problem set #4, due 11/5: do the following exercises from the book: 26, 30, 50 and 52 of 4.1; 6 and 10 of 4.2; and 2, 12 and 36 of 4.3.
• Problem set #5, due 11/12.
• Problem set #6, due 11/19: do the following exercises from the book: 4, 8, 10, 12, 16, 20, 28 and 36 of 6.1.
• Problem set #7, due 11/26.
• Problem set #8, due 12/3.
• Problem set #9, due 12/10.

### Exams

There will be one in-class midterm exam, on Wednesday, November 5. It will cover all material up through and including the lecture on Wednesday, October 29, corresponding to the first four homeworks. See above for the corresponding sections from the textbook.

The final exam is scheduled for Tuesday, December 16, 1:00-3:00 PM, in Art 217. It will cover all material from all 9 homework assignments, corresponding to all lecture material up through the discussion of the Fermat factorization method on 12/5.

The median on the midterm was 57.5%, and the mean was 54%. The letter grade ranges corresponding to the raw scores are as follows:
70-100%: A range
60-70%: B range
40-60%: C range
30-40%: D range
0-30%: F

The median on the final was 54.5%, and the mean was 58%. 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.

### Math Cafe

The Math 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.

### Anonymous Feedback

If you have any feedback on the course, regarding lecture, discussion section, homework, or any other topic, you can provide it anonymously with the below form. Because of the anonymity, I cannot reply to emails from this form, so if you have a question, please just use regular email.

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