General Information
Assignments
Purpose: Math 4400-90 is an online version of our regular
number theory course. Thus it is accessible to those interested in
this beautiful subject who cannot attend the regular class (taught
Fall Semester this year 12:55-1:45). The course is intended for those
with a strong interest in mathematics. Well-prepared, well-motivated
high-school students are encouraged to enroll; they may
register via the High School University Program.
Questions: Contact Jim Carlson (carlson@math.utah.edu)
or Angie Gardiner (gardiner@math.utah.edu), 585-9478.
To register for the on-line number theory
course, sign up for math 4400-90 using
U-Online.
E-mail, communication: After you register, please send
send an e-mail to Jim Carlson at carlson@math.utah.edu
with the subject line "Number theory." I will contact
you one week before the course begins with additional
information. E-mail, the web, and a newsgroup will be
our main form of communication. Occasional meetings at the
University, about once a month, will be scheduled. These
are not mandatory but are recommended.
Text: We will use Joseph Silverman's superb book, "A
Friendly Introduction to Number Theory." It is available at the
University Bookstore and from on-line bookstores (amazon.com, bn.com, etc.).
Assignments, exams: There will be regular weekly
assignments, most of which will be submitted by e-mail. There will be
two scheduled exams. Assignments will be a mix of exploratory
computations, hard problems, proofs, and computer projects.
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What is Number Theory?
Number theory is rich in simple but difficult questions, e.g, the
Fermat Conjecture. Some of these questions are listed below. For
others and more discussion, see the text (Silverman's book).
In
recent years number theory has also become important for its
applications to cryptography and other fields. We will, among other
things, study the famous RSA cryptosystem on which internet security
is based. (See Simon Singh's "Code Book." See also his
"Fermat's Enigma.")
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Why are there infinitely many primes?
How are the primes distributed among the
whole numbers?
Are there infinitely many twin primes
(3-5, 5-7, 11-13, 17-19, etc)? This is an
unsolved question.
What are the integer solutions to the
equation x2 - 301 y2 = 1?
Are there finitely or infinitely many solutions?
Which numbers can be represented as a sum of
squares?
Compute 39988792 mod 9988793.
(RSA requires computations like this).
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