# Math 522: Number Theory

Fall 2003
MWF, 9:00 -9:50.
GMCS 307
San Diego State University

Professor: Mike O'Sullivan
Email: m.osullivan@math.sdsu.edu
Office: GMCS #579, ext. 594-6697
Office Hours: MWF: 10:00-11:30, MWF 1:00-1:45.
Other times: by appointment.

## Text

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

## Detailed Information

 FINAL EXAM INFO SCHEDULE COMPUTER EXPERIMENTS (pdf) ASSIGNMENTS MAPLE WORKSHEETS Basics of algebra in Maple Good primes and primes in arithmetic progression Algebraic integers Modular Arithmetic

## 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 complex 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, Ch. 6 and parts of Ch. 9). Weaved in with these topics will be several applications of congruences: hashing functions (Sec. 5.4), used by computers to store data; check digits (Sec. 5.5), used for passport and ISBN numbers to protect against typographical erorrs; and cryptography (Secs. 8.1-4, 10.2), used for internet security.

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 courses on cryptography (Math 626) and coding theory (Math 525 and 625) and to the Master of Science program in the 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. A day by day schedule (see above) will be maintained to keep you informed of upcoming and past lectures.

 SECTIONS TOPICS TIME §1.1-4 Induction, Fibonacci numbers, divisibility 2 classes §2.1 Base r representations of integers 2 classes §3.1-4,6; §13.1-2 Primes, greatest common divisor, Euclidean algorithm. 6 classes Unique factorization, Diophantine equations. §4.1-3,5 Congruences, the ring Z/n . 5 classes Solving linear and quadratic congruences and linear systems. Chinese remainder theorem. §5.1,3-5 §8.3, §8.4 Applications: divisibility rules, tournament scheduling, 2 classes hash functions, check digits, §6.1, §6.3 Fermat's little theorem, Euler's theorem 2 classes §8.1-4; §10.2 Cryptography: linear cryptosystems, RSA, El Gamal 3 classes §7.1-2; §9.1-4 Multiplicative functions: Euler phi, sum and number of divisors. 5 classes Primitive elements in Z/n .

We will have weekly assignments, three midterms and a final exam. Computer projects (see above) may be done for extra credit up to a total of 100 pts.

 Weekly work 350 Test 1 100 Test 2 100 Test 3 100 Final 350 Total 1000
For the weekly assignments (see above), 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.

For the exams, I will write a list of subject matter and exercises to guide you in your study (link above). These form the material that you are expected to understand upon completion of this course. You can safely ignore material that is not included.

The first exam is tentatively Mon. Oct. 6.

The second exam is tentatively Mon. Nov. 3 .

The second exam is tentatively Mon. Nov. 24 .

The final exam is Fri. Dec. 19, 8:00-10:00.