Florida Atlantic University • Spring 2020 • Markus Schmidmeier Welcome to Number Theory! This course is Introductory Number Theory, MAS 3203, we meet Wednesdays and Fridays, 11:00 a.m. - 12:20 p.m. in General Classroom South 102.

Introductory Number Theory

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.

Starting March 16, FAU is transitioning temporarily to Remote Instruction. For our course this means:

- Classes take place at the usual time as Skype conference calls.
- Quizzes are replaced by homeworks, please submit your solutions as pdf-files via e-mail.
- I'll be available during office hours via Skype.
In case I cannot update this webpage, I will send information regarding homework and final exams via e-mail.

TextbookGareth A. Jones and J. Mary Jones,

Elementary Number Theory,Springer (1998), ISBN 3-540-76197-7.

Topics

- (2-3 weeks, ch. 1-2) Prime numbers are the building blocks for the natural numbers under multiplication. Starting from the Euclidean algorithm, we study prime factorizations, distribution of primes, primality testing and some special primes.
- (2-3 weeks, ch. 3-4) Modular arithmetic is ubiquiteous in everyday life, think of the minutes in the hour, or the last digit on the odometer. The Chinese Remainder Theorem deals with simultaneous congruences. Modulo prime powers, modular arithmetic leads to primality tests.
- (3-4 weeks, ch. 5, 8) We study the Euler function and its use in coding and cryptography. We briefly review the cryptographic system RSA. The Euler function is an example of an arithmetic function, for which the powerful Moebius Inversion Formula holds.
- (2-3 weeks, ch. 9) One of the most famous open problems in mathematics is about the zeroes of the Riemann Zeta Function. It is so important because of its implications for the distribution of prime numbers. Not all is difficult!
- (2-3 weeks) In an application to music theory, numbers are used to build musical scales, and to describe properties of composition. The development of harmony can be traced in the socalled tonal network which has vertices labeled by powers of 2, 3, and 5.

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

Every week there will beHomework:homework assignments.

I will not grade the homeworks, but some problems may come up on the quiz.

Every Friday we will have a quiz of about 20 minutes, about the material covered in class and the homework assignments. The 11 best quizzes count forQuizzes:50%of the grade.

For a presentation of at most 10 minutes, please pick a problem from our textbook which is related to a topic covered in class. The last day for the presentation is April 10. The presentation will count forPresentation:10%of the grade.

The final exam on Wednesday, May 6, 10:30 a.m. - 1:00 p.m. is comprehensive. It will count forFinal Exam:40%of the grade. The first part consists of solutions to three new problems from our textbook of your choice (although one problem may be a previous homework problem, and another problem may be, say, a newspaper problem about numbers). The solutions need to be submitted by Friday, April 24 (Wednesday, April 29 at the latest). The second part is a presentation on Skype during the time of the final exam.Math Learning CenterThere is free math tutoring at the Math Learning Center. Drop-in tutoring is M-R 10-5, F 10-4, U 1-5 in GS207, and M-R 5-9 in Glades Tower Park GPT102 and the Parliament Library.

Further InformationFor 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

W 2-4 and F 2-3 in SE 272, after class if possible, or by appointment.Office hours:

markus@math.fau.eduE-mail:

561-297-0275Telephone:Last modified:by Markus Schmidmeier