Office hours:
M 1011, W 1112, Th 10:3011:30
Office telephone: 642 0648
Fax number: 642 8204
Secretary: 642 5026
email:
ribet@math.berkeley.edu
Textbook
An
Introduction to the Theory of Numbers
by Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery.
You want the fifth edition. Publisher is John Wiley & Sons, Inc.
This is a classic number theory textbook (``Niven & Zuckerman''), updated
by Hugh Montgomery. It is renowned for its excellent problems.
If you think that you've spotted a misprint in the book, first consult
Hugh Montgomery's
list of errors.
For alternative treatments, check out
my guide to
recent and classic books
on number theory.
Some software that may be run on a PC under DOS is available
by ftp
to ftp.math.lsa.umich.edu.
The README
explains how to run the binary executable that you can download from the
server. For more extensive documentation, pick up the files
clint0.pdf,
clint1.pdf,
clint2.pdf,
clint3.pdf,
and
clint4.pdf.
Our book's author Hugh Montgomery wrote to me as follows:
In this set of programs is one that performs powering
congruentially for integers up to 10^{18}.
I wrote the programs in Turbo
Pascal almost 10 years ago. Recently I learned that the Borland Turbo Pascal
compiler has a bug in it that causes programs compiled with it to crash on PCs
running at clock speeds of (roughly) 300MHz or higher. I have installed a
patch in my compiler that fixes this, but I haven't gotten around to
recompiling the programs that are there.
You may prefer to use the
program PARI/gp,
which is now available for a variety of platforms. This program was
developed by and for number theorists, and is used widely in
research. As Monica Chew pointed out to me,
browsable
documentation is available from the
HASSE Server
in Munich.
While I'm at it, I might refer you to
Prime Form,
a primalitytesting program that is available
"for all 32bit Windows operating systems and
has been tested under Windows 95, 98, NT3.51, NT4.0, and Windows 2000."
According to the
General
Catalog,
this course treats
``Divisibility, congruences, numerical functions, theory of primes.
Topics selected: Diophantine analysis, continued fractions,
partitions, quadratic fields, asymptotic distributions, additive problems.''
Examinations
Now that the course is over, you can download the
questions for all three exams in a
single document.
 First Midterm: September 23, in class (questions
and answers).
Here's how the scores were distributed:
X
X X X
X X X X X X X X X X X X
X X X X X X X X X X X X X X X X X X X X X X

5 1 0 1 5 2 0 2 5 3 0 3 5
Median = 20. 5
Average = 20.8
Maximum possible score = 35
 Second Midterm, October 28, in class
(questions
and answers).
Here's how the scores were distributed:
X X
X X X X
X X X X X X X X X X
X X X X X X X X X X X X X X X X X X X X

5 1 0 1 5 2 0 2 5 3 0 3 5
Median = 23. 5
Average = 23.67
Maximum possible score = 35
 Final Exam, Tuesday December 14, 12:303:30, in 2 Evans
(questions
and answers).
Here's how the scores were distributed:
x x x
<x (score=18) x x x x x x x x x x
x x x x x x x x x x x x x x x x x x x x

2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5
Median = 43
Average = 43.74
Maximum possible score = 60
If you're curious about the kind of exams that I have given in the past,
you can check out the Spring, 1998 questions
that I gave in this course.
(The midterm exams were one hour long,
by the way.)
You can even look at the answers to the old
first midterm,
second midterm
and final exam; these documents are
in Adobe
Acrobat format.
I propose to follow the book in order, covering
the first N chapters.
Homework
Homework will be assigned weekly. The grader for this course
is John Voight.
The assignment
will be discussed in class on the day that it is due.
Therefore,
late homework cannot be accepted!
For numerical problems,
the grader encourages you to use computer software as you see
fit. Be sure, however, to include printouts that explain what
you did.

Assignment due September 2:
 § 1.2:
1 (b, d), 3 (b,c), 4b, 9, 13, 14.

Assignment due September 9:
 § 1.2:
24, 25, 46
 § 1.3:
5, 7, 8, 11, 26, 27, 42, 43, 44

Assignment due September 16:
 Suppose that a and b are relatively prime integers for which the
fraction a/b is the nth power of a rational number (n a positive
integer). Suppose that b is positive.
Show that a and b are each perfect nth powers of integers.
 § 1.2:
47, 49, 50
 § 1.3: 51
 § 2.1: 5, 6, 9, 12, 13, 14, 17, 23, 27, 28, 38

Assignment due Tuesday, September 28:
 § 2.1: 40, 41, 43, 44, 47, 48
 § 2.2: 4, 5efg, 6ab, 7, 8, 9

Assignment due Thursday, October 7:
 § 2.2: 10, 14, 15
 § 2.3: 1, 3, 4, 7, 9, 15, 18

Assignment due Thursday, October 14:
 In connection with the RSA algorithm (§ 2.5), work out
numerically what happens if m=391, the exponent k (called e in my
lecture) is 3, and the message to be sent is "123". Calculate the
encrypted message (123)^k mod m and then decrypt this message.
 § 2.5: 2, 3, 4
 § 2.3: 19, 28, 29
 § 2.4: 2, 3 (all parts), 4, 5, 6
Author Hugh Montgomery suggests that
Problem 3 in § 2.4
be worked out using the
programs
that he wrote for IBM compatibles some years ago. For
documentation, see the
parent directory.

Assignment due Tuesday, October 26:
 Study for the second midterm (October 28, in class)
 § 2.4: 7, 9, 10, 13, 14a
 § 2.6: 2, 3, 4

Assignment due Tuesday, November 9:
 § 2.7: 6, 7, 8, 10
 § 2.8: 2, 3, 4, 5, 8ab, 9, 16, 17, 18

Assignment due Tuesday, November 16:
 § 3.1: 8 ace, 14, 15, 19
 § 3.2: 2, 4ace, 6, 7, 10, 11, 13
 Assignment due Thursday, December 2:
 § 3.2: 15
 § 3.3: 4, 7, 8, 9
 § 7.1: 1, 3
 § 7.2: 1
 § 7.3: 2
 § 7.4: 1
 § 7.5: 3, 4
 § 7.6: 1, 2
 § 7.7: 1, 2, 3
 § 7.8: 2, 3, 4, 5, 6
[The problems in § 7.8 should be done
by you in private after the lecture on December 2. They do not need to
be submitted.]
This is a long assignment, so please get started ASAP!
Email Messages
I maintain a mailing list of students in the class. My intention is
to send mail to the entire class when I make significant changes to this
Web page and when I have some news to communicate. You can retrieve
the messages with news in them by clicking on the links below:
Grading
The final course grade will be computed by weighting the exams and
homework roughly as follows: midterm exams, 15% each; homework, 20%;
final exam, 50%. I reserve the right to change the mix at the
end of the semesterone can often see only after the fact how
successful a given exam has been.
I last taught this course three semesters ago. The final
grade distribution in the class was as follows (I neglect +'s and 's):
9 A's, 14 B's, 3 C's, and one
Pass.
Miscellaneous links related to number theory and/or this course
The campus maintains an
official
Web page
for this course, but it's only a skeleton page.
Kenneth A. Ribet
,
Math Department 3840, Berkeley CA 947203840
This page last modified