ROWAN UNIVERSITY

Department of Mathematics

Syllabus for Math 01.503 - Number Theory

Dr Abdul Hassen, Robinson 229E, (856) 256-4500 Ext. 3888, hassen@rowan.edu

Office Hours: M 5:30 – 6:15pm (you are also welcome to just email me questions that you have, make an appointment outside of office hours, or else stop by my office whenever I'm around).

CATALOG DESCRIPTION: This course includes prime numbers, linear and polynomial congruences, the law of quadratic reciprocity, algebraic numbers and integers, other topics in number theory and unsolved problems in number theory.

OBJECTIVES: 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.

TEXT: .....Ivan Niven, H.S. Zuckerman, and H.L. Montgomery, AN INTRODUCTION TO THE THEORY OF NUMBERS, Wiley & Sons, 5th Ed

CONTENT:

1. Basic Concepts: Properties of the integers, Divisibility, Greatest common divisor, Least common multiple, The Euclidean algorithm, Primes, The fundamental theorem of arithmetic

2. Linear Diophantine Equations:  Solution of ax + by = c

3. Congruences: Definition, Arithmetic properties, Linear congruences, Residue classes, Systems of linear congruences and the Chinese Remainder Theorem, Euler's Phi function, Introduction to higher order congruences, Applications: Tests for divisibility useful in arithmetic, Checks for the basic operations of arithmetic     

4. Euler's Theorem: Complete systems of residues, Reduced systems of residues, Euler's and Fermat's Theorems, The exponent to which "a" belongs (mod m)

5. Perfect numbers: The sigma and tau functions, Even perfect numbers

6. Nonlinear Diophantine Equations, The Pythagorean Triples, Fermat's Last Theorem

7. Continued Fractions, Simple continued fractions, finite and infinite, Approximations by convergents, The Pell equation

8. Algebraic Numbers: Polynomials, Algebraic Numbers, Algebraic Number Fields, Algebraic Integers, Quadratic Fields, Units and Primes in Quadratic fields, Unique Factorization

9. Partition Function: Basic properties, Bound for partition function  

Grading policy: Grading is based on homework assignments (40%), a mid-term and a final exam (30%) and student expository paper and presentation (30%).

Letter Grade:            A(-) 85 -100, B(-, +) 70 – 84, C(-, +) 60 – 69, D(-, +) 50- 59, F <49

Homework Policy: There will be homework assignments selected form the text. You must list all exercises assigned on the front cover page of your written solution set.  Circle those exercises that you wrote complete solutions for.

Expository Paper and Student Presentation:

Students are required to write an expository paper and to give a 15-minute in-class presentation (e.g. PowerPoint) on an advanced topic involving Number Theory. Topics must be pre-approved by the instructor.