University of Massachusetts, Amherst
Math 471
Number Theory
Fall 2006
Click here to go the Homework Page
Course News: These will occasionally be posted to the course web site.
For the Friday Review Session, 10-12 am, let's plan to meet in 1114
LGRT (just outside my office). If that room is not available,
we'll try to find another room: in that case,
I'll leave a note on my door about
where we end up in case there are late arrivals.
The class treasure hunt.
The Final Exam is scheduled for Wed. Dec. 20th, 4-6 pm in LGRC
A301. The final exam will cover all the material we discussed during
the term. I have now placed a Sample Final
on the Homework Page. I agreed to give 2 review sessions: one on
Friday Dec. 15, 10-12 (I think) and the other on Monday Dec. 18, 2-4
(I think) ... time/date/place to be confirmed later this week. I will
also try to be available Wed. Dec. 20, 12-2:15 (at the Blue Wall)
and then 2:30-3:30 in my office.
HW 7 and HW 8 will be due Dec. 6 and 13 respectively. There will also be
an extra HW 9 which is not required to be handed in -- those who do
turn it in will boost their homework grade. Details will be announced
in class.
Meeting times: Mon, Wed 12:20-1:35, in Lederle
319.
Instructor:
Dr. Farshid Hajir
Office: Lederle 1118
Phone: 545-6015
Email: hajir@math.umass.edu
Office Hours:
Current office hours are MW 11-12, subject to change.
You are always welcome to set up an appointment by sending me an
e-mail or calling me on the phone.
Text:
Fundamentals of Number Theory by
William J. LeVeque,
Dover Publications,
ISBN 0-486-68906-9 (paperback)
$9.95.
Additionally, I may post my own
course notes
to the course website.
WebCT Bulletin Board: A virtual M471 discussion room will be
set up on your webct account. You can use this space to post messages
related to the course, such as "A group of us are meeting at 9 tonight
in the Science Library to discuss HW 2" or "Does anybody have a clue
what the devil Farshid is asking for in Problem 5 of this week's hw?"
If you get help with assignments from this bulletin board, do not
forget to acknowledge it on your hw (check the hw rules carefully).
Please use this service responsibly. I will monitor it semi-regularly, but
if you want to direct a question specifically at me, the best way to
reach me is by e-mail.
Philosophical
Remarks: They became so numerous, they lobbied successfully for
their own page.
Homework: Homework will be posted on The
Homework Page and collected every Wednesday at the beginning of
lecture. Late homework will not be accepted and the lowest homework
grade will be dropped. Be sure to read and follow the homework
rules.
Attendance: Attendance is required during lectures.
I consider
attendance AND participation important ingredients for your success in
the course. Frequent absences will be reflected in your
grade.
Quizzes:I might give occasional 10-15 minute quizzes in class.
Each of these will count as one homework assignment. There will be
no make-up quizzes.
Exams: There will be two midterms and one final exam.
The midterms will take place during regular class time on
Monday October 16 and Monday November 13. The final will be on
a date to be determined during the regular final-exam period.
Make-up Exam Policy: If you have a legitimate (i.e. multiple
exams at the same exact time, medical problems, emergency absences,
religious observances) scheduling conflict with any of the exams, it
is your responsibility to notify me of the conflict as soon as you
become aware of it. The final decision regarding allowing a make-up
exam is mine, subject to University regulations, of course. Please
note that previously arranged travel plans are not a valid
reason to be given a make-up exam.
Computers: Although this course will not be as computer-intensive as
it has often been in the past, number theory is an inherently computational
subject and we will undoubtedly have some use for computers over the course of
the semester. When the time comes, you will probably be able to get away
with Maple or Mathematica if you're used to those packages. In fact,
however, there is a far better option out there: Pari/GP. Pari/GP has
two main advantages: 1) it is specifically designed for use in number theory;
and 2) it is available at a very reasonable price: free. We'll deal with
this when the time comes, but if you're curious, everything you ever wanted
to know about Pari/GP can be learned at
The PARI homepage.
Extra Credit: Some extra credit problems will occasionally be
included in the homework assignments, or given during class. The number of points for each
problem will vary, as will the difficulty of the problem. The student
with the most points at the end of the semester wins a fabulous prize.
You may hand in Extra Credit solutions at any time throughout the
term, until the last class meeting.
Grading:
homework, quizzes, participation - 30%
2 midterms - 20% each
Final exam - 30%
Grading Scales
|
A
|
>= 93%
|
|
A-
|
>= 90%
|
|
B+
|
>= 86% and < 90%
|
|
B
|
>= 82% and < 86%
|
|
B-
|
>= 78% and < 82%
|
|
C+
|
>=74% and < 78%
|
|
C
|
>= 70% and < 74%
|
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C-
|
>=65% and <70%
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D
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>=60% and < 65%
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F
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below 60%
|
Course topics:
Motivating questions: Pythaogrean triples, Fermat's last theorem,
cryptography.
Divisibility: Divisibility, greatest common divisor, Euclid's
algorithm, linear equations,
Unique Factorization Primes, fundamental theorem of arithmetic.
Congruences: Congruences, Fermat's little theorem, Euler's theorem,
Chinese remainder theorem, powers, successive squaring.
Prime numbers: Counting primes, Mersenne primes, perfect numbers.
Cryptography: RSA cryptography, Carmichael numbers, primality tests.
Primitive roots: Primitive roots, discrete logarithms.
Quadratic reciprocity: Quadratic residues, quadratic reciprocity.
Gaussian integers: Sums of squares, Gaussian integers, arithmetic of
Gaussian integers.
Diophantine approximation: Pell's equation, diophantine approximation.
Additional topics as time permits.