Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Introductory Number Theory

Welcome to Number Theory! This course is Introductory Number Theory, MAS 3203, CNR 20756, we meet Tuesdays and Thursdays, 5:00 - 6:20 p.m. in PS 112.

Aim of this course is to provide a gentle introduction to upper division mathematics. Topic is a very active area in modern mathematics with spectacular applications.

Textbook

George E. Andrews, Number Theory, Dover Publications (1994), ISBN 0-486-68252-8.

Topics

 FundamentalsBasis representation(Chapter 1, 1 week) We review mathematical induction and use it to show the Basis Representation Theorem The Fundamental Theorem of Arithmetic(Ch. 2, 2 weeks) You have probably met the Fundamental Theorem of Arithmetic on several occasions, so the result is no surprise. Here we X-ray the statement: We consider several ways to formulate the result, look into applications, and go through all the details of its proof. Fermat's Little Theorem(Ch. 3, 1 week) The is result is easily shown. But amazingly, it is at the heart of modern public key cryptography! Congruences(Chapter 4 and 5, 3 weeks) We review modular arithmetic and discover that solving linear equations mod n is --- in principle --- similar to solving first order non-homogeneous linear differential equations. Systems of several linear equations, with different moduli ni are dealt with using the famous Chinese Remainder Theorem. Arithmetic functions(Sections 6.1-6.3, 1 week) We consider some multiplicative functions on the natural numbers, a nice example is d(n), the number of divisors of n. SquaresQuadratic residues(Ch. 9, 2 weeks) We will look into an algorithm which decided, for a given number q, which natural numbers are squares modulo q. Sums of two squares(Ch. 11.1, 1 week) Let r2(n) denote the number of ways a natural number n can be written as a sum of two squares. We determine which numbers are sums of two squares, i.e. they satisfy r2(n)>0. Geometric Number TheoryLattice points(Ch. 15, 2 weeks) We use geometry to describe the growth of the two functions mentioned above, d(n) and r2(n).

Objectives

• Learn about a core area in pure mathematics which has spectacular applications and exciting recent developments.
• Practise stating results and conjectures about numbers and discussing applications.
• Learn basic methods of proof writing and practise them in easy examples.
• Apply number theoretic algorithms and discuss their use.
• Select a topic for a presentation, elaborate on it with the help of the instructor, and speak in front of the class.

Credit

Homework:   Every week there will be homework assignments. I will not grade the homework assignments, but some problems may come up on the weekly quiz.

Quizzes:  Every Thursday we will have a quiz of about 20 minutes, about the material covered in class and the homework assignments. The ten best quizzes count for 30% of the grade.

Midterm Exam:   The midterm exam will count for 15% of your grade. It is scheduled for Thursday, February 21.

Presentation:   For every student, a presentation of about 15 minutes, typically about one of the textbook problems, will count for 15% of the grade. (If enrollment is large, we may have to omit the presentations.)

Final Exam:   The final exam on Thursday, April 25, 4:00 - 6:30 p.m., is comprehensive.  It will count for  40% of your grade.

Further Information

For the Disability Policy, the Make-Up Policy, the Code of Academic Integrity, Religious Accommodation, my Grading Scale and Financial Assistance Opportunities please visit Infos for all my courses.

Contact Me

Office hours:  Tuesdays and Thursdays, 10-12 a.m. in SE 230, or after class.

E-mail:   markus@math.fau.edu

Telephone:   561-297-0275