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
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:
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 upperdivision 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:
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 911 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. 35):
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:
 Assignment due Friday, September 22:
 § 1.24 4649
 § 1.25 50, 51, 52, 53
 § 1.27 57 (parts fo)
 Assignment due Friday, September 29:
Exercises 6164 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 (p1) 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_{p1} is divisible
by p. For example,
F_{10} = 55 is divisible by 11, and
F_{28}= 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 58PM, 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.
You can look at the
Math 54 group
from last fall to see some ways in which our group might be used.
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