Math 522: Number Theory

Course number: 21475
Fall 2005
MWF, 12:00 -12:50.
GMCS 325
San Diego State University
Final Exam: Wed. Dec. 21, 10:30-12:30

Professor: Mike O'Sullivan
Office: GMCS #579, ext. 594-6697
Office Hours: M: 1:00-1:30, Tu: 2:00-3:30, W: 1-1:30, 2:15-3:30, Th: 2:00-3:30.
                             Other times: by appointment.


Rosen, Elementary Number Theory and Its Applications 5th ed. (This is a new edition.)

Detailed Information

My tutorials page for Maple and Magma.

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 considered to be one of the purest and most beautiful areas of mathematics. Over 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.



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.

§1.1-4 Induction, Fibonacci numbers, divisibility 4 classes
§2.1 Base r representations of integers 2 classes
§3.1-4,6; §13.1-2 Primes, greatest common divisor, Euclidean algorithm. 8 classes
Unique factorization, Diophantine equations.
§4.1-3,5 Congruences, the ring Z/n . 7 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, 3 classes
hash functions, check digits,
§6.1, §6.3 Fermat's little theorem, Euler's theorem 3 classes
§8.1-4; §10.2 Cryptography: linear cryptosystems, RSA, El Gamal 4 classes
§7.1-2; §9.1-4 Multiplicative functions: Euler phi, sum and number of divisors. 5 classes
Primitive elements in Z/n .


There will be 10 problem set assignments, two midterms, a computer project and a final exam.

Written assignments should be carefully and neatly presented. Computer exercises should have explanatory comments and be well organized.

The relative weights of the work are given below.

Weekly work 250
Computer Project 100
Tests 300
Final 350
Total 1000

I dont assign assign

For undergraduates an A is above 800, B above 700, C above 600.

For graduate students an A is above 850, B above 750, C above 650.