Ma458 Number Theory
Math majors, both those planning on graduate school and those
planning to teach high school.
Computer Science majors looking for a theoretical elective. High
school math teachers. Anyone who likes mathematics and wants to learn
Goals of This Course
To learn some number theory, of course. To learn why you might want to
learn some number theory (if you don't already know). To enjoy
learning number theory. To practice reading and writing mathematics.
To stretch yourself intellectually.
Course structure, grades
I expect you to pay attention in class, to do the homework, to read
about number theory in different places
and to ask questions about things
you do not understand.
There will be lots of homework.
If this is your first course beyond calculus and linear algebra
you may find the problems different from those you have encountered
before in a math course. Few of them have what one could narrowly call
an answer. All require playing with numbers, observing patterns,
writing convincing arguments.
There will be a midterm and a term paper. The final examination will
probably be (mostly) take-home.
I am considering structuring a significant fraction of the semester
as a seminar. If I do, each student will present. All will be
responsible for all the material covered.
I do not use an algorithm to compute grades. Here's roughly how I
weigh the factors:
I know that the students in the class are starting from different
places mathematically. I expect each of you to learn at a level that's
appropriate. So the same score on an exam may mean an A for someone
and a B for someone else.
- 60% homework (including seminar presentation, final paper)
- 25% exams
- 15% enthusiasm, class participation, effort expended
I do not grade on a curve. Everyone can earn an A. (Wouldn't that be a
TTh 11-12:15, Wheatley W02-0124
Getting help and encouragement
I am always glad to talk about mathematics. Whenever
my door is open (most of TTh) you are free to
ask questions. Email to
eb at cs dot umb dot edu
is the best way to reach me, both
during the day and evenings and weekends. My office phone is 287-6444
but I don't have voice mail and I.m not always there. My home phone is
(617) 969-2892, but please use that only for real
emergencies. (Emergencies that happen after 8 PM should wait for the
Often other students in the class can help you. It's worth learning
how to work with, learn from (and help) your peers - how else will you
keep learning when you've left school?
The text for this course is
A Friendly Introduction to Number Theory. The first six chapters
are available on line at that site, which will give you time to get
Here are several others you may find
might find useful. The first few are inexpensive and I recommend them
- Richard Friedberg,
Adventurer.s Guide to Number Theory, Dover, ISBN: 0486281337,
You can read the reviews on
but buy it from the publisher, or the bookstore.
- My text: Ethan Bolker,
Theory: An Algebraic Approach,
Dover, ISBN: 0486458075, $13.95.
This treatment is designed for students who have studied abstract algebra.
I earn no royalties on the sale, so it's OK to recommend it.
It too is so inexpensive that you should order direct from Dover.
The Amazon savings are almost infinitesimal, unless you're getting
the Theory of Numbers is also available from Dover, for
$14.95. You can find used copies on the web
(try Abe books) for a dollar
or two. Read the
Use of Computers
None is required. Some of the topics we touch on have applications in
computer science, and we may mention those applications, particularly
when we learn about cryptography. But number theory is an old
discipline, and this is an old fashioned course.
No technology. All you need is pencil and paper -
perhaps a calculator if you're not fast at arithmetic. But there are
infinitely many integers and almost all of them are larger than
your calculator can handle, so you can't substitute calculation for
That said, you may find that automating some computations helps you
understand the number theory, both because it gives you access to
larger examples than you can construct by hand, and because to code
an algorithm you must master it. In the past students have written
programs in C, Pascal, Scheme, mathematica and for their programmable
If you want to use a computer and don't want to write your own
programs, there is lots of number theory software out there.
Most of it is for professionals but there are packages that are
essentially instructional. I don't know of any, but would be pleased
to help you decide whether one you found was worth working with.
Section 504 of the Americans with Disabilities Act of 1990 offers
guidelines for curriculum modifications and adaptations for students
with documented disabilities. If applicable, students may obtain
adaptation recommendations from the
Ross Center for Disability
Services, M-1-401, (617-287-7430). The student must present these
recommendations and discuss them with each professor within a
reasonable period, preferably by the end of Drop/Add period.
Students are required to adhere to the University Policy on Academic
Standards and Cheating, to the University Statement on Plagiarism and
the Documentation of Written Work, and to the
Code of Student Conduct
as delineated in the catalog of Undergraduate Programs, pp. 44-45, and
48-52. The Code is available online at
For my own elaboration on the rules concerning the acknowledgment of
intellectual debt, particularly appropriate in this course where the
focus is on teamwork, see