## matthias beck

some of matt's pictures
 MATH 310 Elementary Number Theory Fall 2018

 Lecture Mon/Wed/Fri 2:00-2:50 p.m. TH 404 Prerequisites: MATH 301 with a grade of C or better Instructor Dr. Matthias Beck Office Thornton Hall 933 Office hours Mon 11-12, Wed 10-11, Fri 1-2 & by appointment

Course objectives. Number Theory studies the integers: numbers like 0, 1, -3, 34, ... which we used since childhood. A fundamental concept is that of divisibility: the integer a divides the integer b if we can find an integer c such that b=ac. This simple concept gives rise to a beautiful theory, encryptions schemes which are used on any computer today, and many famous open problems in mathematics, among other things.

As an example, we will consider prime numbers: those integers >1 that are only divisible by themselves and 1. There are infinitely many prime numbers--one of the many theorems we will prove in this course. The only even prime is 2, all others are odd. To say this in a more sophisticated way: there are infinitely many primes that have a remainder of 1 when divided by 2. Can one say something similar when we divide primes by 3? The only prime that does not give a remainder when divided by 3 is 3 itself. So all others give a remainder of 1 or 2. Are there infinitely many primes in both cases? To give an open problem, we mention twin primes: those are pairs of primes that differ by 2. Are there infinitely many twin primes?

Syllabus. Topics in this course will include:

• Divisibility
• Primes
• Congruences
• Arithmetic functions
• Primitive roots
• Continued fractions

Textbooks.

Participation. Most of the material covered in this class will be worked out in small groups during class sessions; this philosophy is sometimes called inquiry-based learning. It will thus be essential that every student participates actively in every class. If you have to miss a class due to a medical or family emergency, please let me know before the class; otherwise, I expect you to be in class and actively engaged.

Homework. The worksheets we will be working on in class contain more material than what can be covered in a normal class session. I will assign certain problems from each worksheet as homework problems. The worksheets will be due before on the following Tuesday. You may hand them in early to be able to correct your mistakes. Although you may (and should) work together with your class mates, the solutions you hand in have to be your own.

Sage. You will be expected to use the open math software sage in some of your homework assignments. Here is a good introduction to sage.