Number Theory

Instructor: Christopher Sadowski

Email: csadowski at ursinus dot edu

Office: Pfahler Hall 101E

Office Hours: MW 1:30-3:00, TTh 1:00-2:00

Text: Elementary Number Theory and Its Applications, Kenneth H. Rosen, Sixth Edition

Course Objectives: Students will learn the basic ideas and proof techniques in elementary number theory, as well as the role of number theory in the history of mathematics. Students will be required to know:

Prime numbers, prime factorization, greatest common divisors, and the fundamental theorem of arithmetic
Solve Diophantine equations and linear and polynomial congruences
Euler's phi-function, Fermat's Little theorem, Wilson's Theorem, and Euler's Theorem
Mobius Inversion
Primitive roots modulo prime powers

Learning Goals: This course will meet the following departmental learning goals:

Organize and synthesize evidence to identify patterns and formulate conjectures
Demonstrate mastery of the standard proof techniques
Solve problems with mathematical components, and use standard software packages when appropriate
Communicate to technical and non-technical audiences, and work independently and in groups

Location: Pfahler Hall 007

Meeting times: Monday, Wednesday, Friday 10:00-10:50

This is a 4 credit course that meets for 3 hours weekly. The extra hour of work will be made up with homework assignments, a final project, and in-class presentations.

Grading:

-30% Homework

-10% Midterm 1

-10% Midterm 2

-10% Midterm 3

-15% Final Project

-25% Final Exam

Homework: Homework will be sent out via email and posted on Canvas. Students will be given a small amount of extra credit on each assignment for homework typed up using LaTeX. Proofs are expected to be fully rigorous, and you should always state your assumptions and which theorems you are using.

Final Project: Students will be expected to complete a final project on a topic of their interest, after consultation with the professor. Students will also give a short, 10 minute presentation on their topic. A list of possible topics can be found here, but feel free to explore a topic of your own interest!

Exams: The following dates are tentative:

Exam 1: 9/25

Exam 2: 10/27

Exam 3: 11/27

Final Exam: TBA

Academic Honesty: Students may work together and discuss assignments with one another, but all work handed in must be solely the student's. Any incident of cheating on a quiz or exam will results in a grade of 0 for the assignment, with no make-up allowed. A second incident of cheating will result in a failing grade for the course. All incidents of cheating will be reported to the Dean's Office. Please refer to the Student Handbook on Academic Honesty and the Statement on Plagiarism.

Attendance: Students are expected to attend all classes. If you are unable to attend class for a legitimate reason, please email me before class.

Inclement Weather Policy: Students will be emailed in the event that class is cancelled due to inclement weather.Ac

Accommodations Policy: Students requiring accommodations should provide me with the appropriate paperwork from the Accommodations Office at the beginning of the semester.

SPTQ: Towards the end of the semester, students will be reminded to fill out SPTQ forms. These forms are invaluable to both the instructor and the department, since they provide valuable feedback to the instructor on how to improve the course, and provide evaluation information to the department chair. Honest constructive feedback (both positive and negative!) is appreciated.

Inclusive climate in the classroom: In this class we will work to promote an environment where everyone feels safe and welcome, even during uncomfortable conversations. Every voice in the classroom has something of value to contribute to class discussion. Because the class will represent a diversity of individual beliefs, backgrounds, and experiences, every member of this class must show respect for every other member of this class. You are encouraged to not only take advantage of opportunities to express your own ideas, but also, learn from the information and ideas shared by other students.

Syllabus (subject to change as the course progresses):

Topics we will cover include:

1.2 Sums and Products

1.3 Mathematical Induction

1.4 Fibonacci Numbers

1.5 Divisibility

3.1 Prime Numbers

3.2 The Distribution of the Primes

3.3 Greatest Common Divisors and their Properties

3.4 The Euclidean Algorithm

3.7 Linear Diophantine Equations

3.5 The Fundamental Theorem of Arithmetic

4.1 Introduction to Congruences

4.2 Linear Congruences

4.3 The Chinese Remainder Theorem

6.1 Wilson's Theorem and Fermat's Little Theorem

6.2 Pseudoprimes

6.3 Euler's Theorem

7.1 The Euler Phi-Function

7.2 The Sum and Number of Divisors

7.3 Perfect Numbers and Mersenne Primes

7.4 Mobius Inversion

8.1 Character Ciphers

8.2 Block and Stream Ciphers

RSA cryptosystem

9.1 The Order of an Integer and Primitive Roots

9.2 Primitive Roots for primes

9.3 The Existence of Primitive Roots