**Number Theory**

**Math 522**

Fall 2001

Meeting: MWF, 11:00 - 11:50 am.

Bus. Adm./Math Building #260

San Diego State University

**Professor**: Mike O'Sullivan

**Email**: m.osullivan@math.sdsu.edu

**Office**: Bus. Adm./Math Building #217, ext. 594-6697

**Office Hours**: MWF: 10:00-10:30, 12:00-2:00.

Other times: by appointment or good fortune (I will normally be in my
office and available).

## Text

Rosen, *Elementary Number Theory
and Its Applications* 4th ed.

## Schedule and Assignments

## Course Description

Number theory is one of the oldest and richest subjects in
mathematics. One aspect of the subject that delights mathematicians
is that a problem that is very easy to pose can require very difficult
and profound mathematical structures to solve.
Number theory is also considered
to be one of the purest and most beautiful areas of mathematics. Yet
in the last few decades it has become an important applied subject as
well; for example, the security of internet communications depends on
an application of number theory.
We are using an excellent text book in this course. It gives a good
introduction to the fundamentals of number theory, includes several practical
applications and has interesting discussions of
some important unsolved problems and historical topics.
The core material of the course is primes and divisors (Ch. 3) and
congruences (Ch. 4 and Ch. 6).
Weaved in with these topics will be several applications of
congruences: hashing functions (Sec 5.4), used by computesrs to store
data; check digits (Sec 5.5), used for passport and ISBN numbers to
protect against typographical erorrs; and RSA cryptography (Sec 8.4),
used for internet security.
We will also look at Mersenne and Fermat numbers and the worldwide
collaborative effort to factor them (you can help!). If time permits,
we will cover continued fractions which have an interesting
relationship with the Euclidean algorithm and Fibonacci numbers and
are used in factoring algorithms and approximation of real numbers.
I hope that you will appreciate the beauty of number theory and
acquire a taste for pursuing applications of the subject. I'd like to
entice you into the cryptography course (Math 626) offered this Spring
or to the new Master of Science program, Mathematical Theory of
Communication Systems.
## Schedule

Here is a rough idea of the amount of time I
expect to spend on each topic. I am also open to suggestions if the
class would like to spend more time on certain topics or cover items
not listed here.
Here is a rough idea of the amount of time I
expect to spend on each topic. I am also open to suggestions if the
class would like to spend more time on certain topics or cover items
not listed here.
## Grading

We will have weekly assignments, two midterms and a final exam.
For the weekly assignments, there will be a small number of problems
(10 or so) which you should write up carefully. I will either collect
these and grade them or give a short quiz with some selection of the
problems.

There will be a much larger number of problems assigned to do, but not
to write up formally. These form the material that you are expected to
understand upon completion of this course. You can safely ignore the
problems that are not assigned. The midterms and the final exam will
be based on the material in the assigned problems, but will not
necessarily be identical to something assigned.

Weekly work |
350 |

Test 1 |
150 |

Test 2 |
150 |

Final |
350 |

Total |
1000 |

The first exam is TBA.

The second exam is TBA.

## Assignments