Syllabus for Number Theory, Dr. Ellen Gethner

Applied Number Theory: CSC 4110/5110 (Spring 2020)

Professor Ellen Gethner

Syllabus

automatically updated on 16 January 2020

[ Instructor and Office Hours | Class Time and Room | Textbook and Software | Prerequisites | Objectives | Grades and Policies | Schedule| Academic Deadlines | ]

Instructor

Dr. Ellen Gethner
Email: ellen dot gethner at ucdenver dot edu
Office: Lawrence Street Center LW 817
Phone: 303 315 1405
Office hours:

Tuesday and Thursday 11:30-12:15pm
Tuesday and Thursday 3:30-5:00pm

Class Time and Room

Tuesdays and Thursdays 2:00-3:15pm in Lawrence Street Center 836

Required Textbook and Software and Laptop

Textbook: Elementary Number Theory, 6th edition by Kenneth Rosen, published by Pearson. Available at the Auraria book store and many other places.
Software Requirement: All class notes and homework assignments are given as Mathematica notebooks; you can get Mathematica free through the University of Colorado HERE. Be sure to use your official university email address when you work through the steps to get the software.
Laptop Requirement: You must bring your laptop (with Mathematica installed) to class on class lecture days (not needed on Exam and Quiz days).

Other Resources (send interesting links and I'll add them)

introductory prime number theory resources
Riemann Hypothesis
A page with free number theory textbook downloads
RSA Challenge (archived)
The Number Theory Web
Number Theory Visualizatons
Gaussian Primes What does the graphic at the top of this page mean?

Prerequisites

CSCI 2411 (Discrete Structures)

Course Objectives

We will begin by acquiring knowledge of the basic tools and concepts in Number Theory such as Integers, Primes, Divisibility, GCD, Congruences, Wilson's and Fermat's Theorems, Pseudoprimes, and Multiplicative Functions (such as the Euler phi funcion). Then we will move on to more advanced topics such as Cryptology and Computer Security, Quadratic Residues, Continued Fractions, Diophantine Equations, and the Gaussian Primes. We will study computational and algorithmic aspects of each topic as appropriate.

Grades and Policies

Assignments and Quizzes: Homework assignments (6) will count toward 40% percent of your grade. You will not hand in homework assignments: each assignment will have an associated in-class quiz (30 minutes), which will be given only once. Homework 6 material will be tested on the final exam (along with other material). The lowest score will be dropped.
- Each assigment will have an assortment of straightforward problems as well as challenging problems.
- To prepare for the quizzes, write up your ideas on all of the problems that you have given some thought to, even if you do not believe you have a complete solution.
- Be sure to be neat, write in complete sentences, and do not leave out any steps: clarity of exposition may contribute toward a positive score. Please talk to one another about the assignments, but the write-ups should be done individually so that you are sure to be prepared for the quiz.
Make-up quizzes: None. I will drop your lowest score. There are no exceptions.
Midterm Exam (30% for undergrads, 20% for grads) Given once only during class on Thursday 12 March.
Final Exam (30% for undergrads, 20% for grads) Given once only during class on Tuesday 5 May.
CS Graduate Students will give a research paper presentation and software demonstration (20% of final grade) . 60 minute research presentation to be given to your colleagues in class. Early in the semester I will hand out a research guidelines document. You are expected to find, read and thoroughly understand the (peer-reviewed legitimate and substantive) paper and the contents of the bibliography. In addition you will implement the ideas about which you have learned.You will prepare and present a conference-style presentation on your paper, complete with citations and a software demonstration at the end of the semester. A guide will be posted with the list of possible papers with more explanation.

Schedule

Lecture	Date	Topic(s)	Comments	Reading	Assignment, Quiz, and Exam Schedule
One	21 and 23 January	Introduction to Number Theory	Vocabulary and Definitions; Mathematica demo	Chapter 1.1-1.5, 2.1-2.3 and class notes	Homework problems are in this week's mathematica notebook
Two	28 and 30 January	Vocabulary and Definitions, continued	Primes and greatest common divisor	Chapter 3.1 and 3.2. Class notes.
Three	4 and 6 February	GCD, Euclidean Algorithm, Fundamental Theorem of Arithmetic		Chapter 3.3, 3.4, and 3.5 and class notes	Quiz 1 on Thursday; Homework problems are contained in the mathematica notebook
Four	11 and 13 February	Fermat Numbers, Congruences, Chinese Remainder Theorem	Homework problems are contained in the mathematica notebook	3.6, 4.1, 4.2, 4.3 and class notes
Five	18 and 20 Feb	Wilson's Theorem, Fermat's Little Theorem, Pollard Factoring Method		6.1 and class notes	Quiz 2 on Thursday; Homework problems are in this week's mathematica notebook
Six	25 and 27 Feb	Pseudoprimes, Euler's Theorem (generalizes Fermat's Little Theorem), Euler's phi function		6.2, 6.3, 7.1 through example 7.4	Quiz 3 is on Thursday and covers material through 6.1; The next batch of homework problems are in this week's mathematica notebook
Seven	3 and 5 March	Perfect Numbers, Mersenne Primes, Cryptography	Main topic on Thursday is public key cryptography, particularly RSA	8.1, 8.3 (not covered in class, but useful to know), 8.4 (RSA) and class notes
Eight	10 and 12 March	Midterm (Thursday) plus the Diffie-Hellman Key Exchange, Playing Poker over the phone, Secret Sharing Schemes (Tuesday)	read 8.6 for Tuesday and exam one will be on Thursday 12th March		The midterm will cover material up through 7.3 in the Mathematica notebook "NumberTheoryWeek7_Part1"
Nine	17 and 19 March	Another Threshold Scheme, Primitive Roots, Discrete Log Problem, and ElGammal Encryption	Alternative approaches to these topics are given in this week's Mathematica notebook	9.1-9.4 and class notes
null	23-27 March	Spring Break: no classes
Ten	31 March and 2 April	Topic	Special Topic	Guest speaker: TBA	Quiz 4 on Thursday;
Eleven	7 and 9 April	Elliptic Curve Cryptography		Class notes
Twelve	14 and 16 April	Quadratic reciprocity		9.4 (again), 11.2, 11.2, and class notes	Quiz 5 on Thursday;
Thirteen	21 and 23 April	TBA	Grad student presentations	Lecture slides
Fourteen	28 and 30 April	TBA	Grad student presentations	Lecture slides
Fifteen	5 and 7 May	Tuesday EXAM	Exam plus research talk	Lecture Slides	Tuesday 5th May, Exam 2. Exam 2 is comprehensive and will include material from the week 12 Mathematica notebook on quadratic reciprocity.
Sixteen	Week of 11 May	Monday office hours to be determined.