Mathematics 115

Fall 2006
MWF 2:10-3PM, 60 Evans Hall

Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.

CourseWeb Math 115 page: Enrollment information

Professor Kenneth A. Ribet

email:

Telephone: 510 642 0648

Fax: 510 642 8204

Office hours (885 Evans Hall)

photo of Ribet giving a lecture
to high school students at the MSRI in Berkeley

Textbook

Topics in the theory of numbers by Erdös and Suranyi. (Purchase it from Amazon, Springer or a local bookstore.)

The Erdös text emerged from a field of four proposed candidates. The others were:

An introduction to the theory of numbers by Niven, Zuckerman and Montgomery. This is an excellent book with outstanding exercises. However, it has been used frequently in Math 115, and I was eager for a change. It is also more than twice the price of our textbook. (See Textbook Tempest: Students and Professors Decry Price Surges in the August, 2006 issue of the Notices of the American Mathematical Society for a discussion of mathematics textbook pricing.)
A computational introduction to number theory and algebra by Victor Shoup.
Elementary number theory by William Stein, which was used by Walter Kim over the summer.

Thanks for your comments on the four books.

Syllabus and Course Description

This course is intented to be elementary in the sense that no other upper-division course is needed as a prerequisite. Our book is indeed quite elementary and concrete. On the other hand, it requires readers' close attention and includes some exercises for which cleverness and insight may be helpful. Even its proofs of standard results are often quite novel. Unlike many recent books, it does not emphasize elliptic curves or cryptography. It seems to avoid mention of continued fractions. We will follow the book closely, omitting Chapter 4 if we can. Here is a brief description of what we might study this semester:

Chapter 1: Fundamental properties of the set of integers, including prime factorization and common divisors.
Chapter 2: Congruences, including quadratic reciprocity.
Chapter 3: Approximation of real numbers by rational numbers.
Chapter 5: Properties of prime numbers, including Erdös's proof of Bertrand's Postulate.
Chapters 6-7: Sequences of integers and Diophantine problems.

Graduate Student Instructor

Aaron Greicius of the math department will be participating in this course in various ways: he'll be grading homework, holding office hours, assisting with exam grading and taking over occasional lectures. His office hours: Wednesday 9-11 in 834 Evans.

Homework

Homework assignments will be due on Wednesday each week. There will be a short set of problems for the first week.

Assignment due September 1:
- § 1.2 (i.e., pp. 3-5): 6, 11, 12 [These problems can be seen from Google books by searching for "Erdos Suranyi".]
- § 1.3: 14 [This problem can be seen from an independent scan.]
Assignment due September 8:
- § 1.5: 17, 18
- § 1.6: 19, 20
- § 1.7: 21
- § 1.8: 25
- § 1.12: 27, 28
- § 1.14: 34, 35
Problem 34 is equivalent to a noterious job interview question: A student comes into a locker room and opens all the lockers. A second student closes every second locker door, beginning with the second one. A third student changes the status of every third door, beginning with the third. (She closes the third door but opens the sixth, which was closed by the second student.) A fourth student changes every fourth door, beginning with the fourth. This goes on until N students have gone through; N is the number of lockers. At the end of the process, which lockers are open?
Assignment due September 13:
- § 1.20 39-44
Assignment due Friday, September 22:
- § 1.24 46-49
- § 1.25 50, 51, 52, 53
- § 1.27 57 (parts f-o)
Assignment due Friday, September 29: Exercises 61-64 on page 36. Exercise 65 is interesting and is recommended; consider it as an optional problem.
Assignment due Friday, October 6:
- § 2.6 3, 4, 5
- § 2.11 11, 12, 13
- § 2.17 15, 16
Assignment due Friday the 13th:
- § 2.22 21
- § 2.35 22, 23, 24, 25, 26
- § 2.36 27
Assignment due Friday the 20th:
- If p is a prime > 3, show that sum of the reciprocals of the first (p-1) positive integers, when written as a fraction, has a numerator divisible by p^2. For example, 1 + 1/2 + ... + 1/10 = 7381/2520 and 7381= 121*61. [Hint: you've essentially done this problem on a previous assignment!]
- § 2.38 30, 31
- Use Euler's lemma (p. 75) to decide whether 13 is a quadratic residue modulo 257. (Compare the discussion in § 43.)
- § 2.51 32, 38 (OK to use Th. 2.24 if you need to)
Assignment due Friday, October 27: Chapter 3, problems 4, 5, 6, 7
Assignment due Monday, November 6: Chapter 5, problems 1, 3, 4, 6
Assignment due Monday, November 13: Chapter 6, problems 1, 5 and 6 (both basically done in class, so don't hand in), 7, 8, 9, 10. Also, suppose that 5 is a square mod p, where p is a prime. Show that the Fibonacci number F_p-1 is divisible by p. For example, F₁₀ = 55 is divisible by 11, and F₂₈= 317811 is 10959 times 29. (Never mind the last problem; I did it in class.)
Assignment due Monday, November 20
Assignment due Friday, December 1 [Added November 30: a hint for the last problem.]
Semester's last assignment, due Friday, Decmember 8

Grading and Examinations

Course grades will be based on a composite "total score" that is intended to weight the course components roughly as follows: midterm exams 15% each, homework 25%, final exam 45%. Please note the following exam dates and times:

First midterm exam, Wednesday, September 27, 2006, in class. [Questions and possible solutions, written by Ribet.]
Last midterm exam, Wednesday, November 1, 2006, in class. [Questions and possible solutions, written by Ribet.]
Final examination, Saturday 5-8PM, December 16, 2006 in 60 Evans; happy holidays! [Questions and possible solutions, written by Ribet.] Results: This was not your greatest showing, friends. The median score was 26 out of 45. There were only two scores over 40; scores ranged from 10 to 45.

Previous math 115 courses taught by Ribet:

See also Richard Borcherds's archive of web pages for Berkeley math courses.

Online discussion

Please register for google groups and join the group Math 115.

Google Groups UCBMath115

Browse Archives at groups.google.com

You can look at the Math 54 group from last fall to see some ways in which our group might be used.

Last Updated: