Department of Mathematics


Math 01.503 - Number Theory


Math 01.503  Number Theory                                                                                           3 S.H.

This course includes prime numbers, linear and polynomial congruences, law of quadratic reciprocity, algebraic numbers and integers, other topics in number theory and unsolved problems in number theory. This course may not be offered annually.


It is the purpose of this course to present to the student an introduction to an area of pure mathematics which, although it does not abound in practical application, has intrigued many non-professionals people, as well as the greatest mathematicians from the time of the ancient Greeks to the present.


            1.         Basic Concepts

                        1.1       Properties of the integers

                        1.2       Divisibility: definition and the division algorithm

                        1.3       Greatest common divisor

                        1.4       Least common multiple

                        1.5       The Euclidean algorithm

                        1.6       Primes

                        1.7       The fundamental theorem of arithmetic

            2.         Linear Diophanite Equations

                        2.1       Solution of ax + by = c

........... 3.                  Congruences

                        3.1       Definition

                        3.2       Arithmetic properties

                        3.3       The linear congruences ax = b (mod m)

                        3.4       Residue classes

                        3.5       Systems of linear congruences and the Chinese remainder theorem

                        3.6       Euler's Phi function

                        3.7       Introduction to higher order congruences


3.8       Applications:

                        3.8.1    Tests for divisibility useful in arithmetic

                        3.8.2    Checks for the basic operations of arithmetic    

            4.         Euler's Theorem

                        4.1       Complete systems of residues

                        4.2       Reduced systems of residues

                        4.3       Euler's and Fermat's theorems

                        4.4       The exponent to which "a" belongs (mod m)

            5.         Perfect numbers

                        5.1       The sigma and tau functions

                        5.2       Even perfect numbers

            6.         Nonlinear Diophantine Equations

                        6.1       The Pythagorean problems

                        6.2       Fermat's last theorem

            7.         Continued Fractions

                        7.1       Simple continued fractions, finite and infinite

                        7.2       The closeness of approximations by convergents

                        7.3       The Pell equation

            8.         Applications

                        8.1       Cryptology

                        8.2       The calendar problem


            The following are indicative of texts suitable for the course:

.....Ivan Niven, H.s.< Zuckerman & H.L. Montgomery, AN INTRODUCTION TO THE THEORY OF NUMBERS, Wiley & Sons, 5th Ed

Burton, David, ELEMENTARY NUMBER THEORY, W. C. Brown Publishing Company, Dubuque, IA, 1989.

Niven, I. and Zuckerman, H., THEORY OF NUMBERS, John Wiley Publishing Company, NYC, 1990.

Rosen, Kenneth, ELEMENTARY NUMBER THEORY, Addison-Wesley       Publishing Company, Reading, MA, 1988.

Rev.: 5/00