# MTHT 435 Foundations of Number Theory

## Fall 2005

Instructor: David Marker
Class Meets: MWF 12:00 303 Adams Hall
Office: 411 SEO
Office Hours: M,W: 11-12, F:8:30-10:00
phone: (312) 996-3069
e-mail: marker@math.uic.edu

### Prerequisites

Grade of C or better in MATH 215, or permission of instructor.

### Description

A first course in number theory. The natural numbers and the integers have fascinated mathematicians for centuries.
• Mathematics is the Queen of Science, and Arithmetic the Queen of Mathematics. -- C. F. Gauss
• God made the Integers, all the rest is the work of man. -- L. Kronecker
• No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years -- G.H. Hardy
Number theory has an endless suply of challenging problems that are simple to state. The ancient Greeks knew there were infinitely many prime numbers and knew how to find all the infinitely many integer solutions to the equation
X^2+Y^2=Z^2

Yet, for centuries mathematicians labored to prove that if \$n>2\$, there are no solutions to
X^n+Y^n=Z^n

in the nonzero integers, with Andrew Wiles succeding only in the 1990s. Other problems, like the Twin Prime Conjecture that asserts that there are infinitely many prime numbers p where p+2 is also prime, are still open.

G. H. Hardy would be surprised to learn that number theory is now studied not only for it beauty, but also for its applications. Number theoretic methods play a key role in cryptography including internet data encryption.

In this course we will study the basic properties of the arithmetic of the integers: divisibility, primality, congruences, quadratic residues, sums of squares and diophantine equations. We will also look at some of the applications of number theory in cryptography.

### Texts

Required: G. Jones and J. Jones, Elementary Number Theory, Springer 1998.
Supplementary: J. Silverman, A Friendly Introduction to Number Theory, Prentice Hall 2001.

Jones and Jones is a straightforward basic text in number theory. We will cover most of chapters 1-7,10,11. This roughly corresponds to chapters 1-27 of Silverman. Silverman is a more informal text that also gives glimpses of many advanced topics in number theory. Some of you might find it useful or inspiring. I will probably briefly cover a few topics that are in Silverman but not Jones and Jones.

#### Homework

Jones and Jones provides solutions for all exercises in the book. Each week I will select some of these and encourage you to do them and check your answers to make sure you understand the basic material. There will also be weekly problem sets. These will be a mix of basic and more computational problems (and perhaps ocassionaly optional programming problems). The problem sets will be graded. The two lowest problem set grades will be dropped.

There will be 2 midterm exams and a final exam. Each midterm will count for 25% of your final grade. The final will count for 35% and the problem sets will count for 15%.
Midterm 1: Friday September 30
Midterm 2: Friday November 11
Final Exam: Friday December 9, 8:00 am

#### Suggested Practice Problems

The following practive problems are from Jones & Jones. Solutions are in the back of the book.
• Chapter 1: Exercises: 1.2, 1.3, 1.4, 1.5, 1.7--1.16
• Chapter 2: Exercises: 2.1, 2.2, 2.3, 2.6, 2.7, 2.11--2.14, 2.17, 2.18
• Chapter 3: Exercises: 3.1, 3.2, 3.4--3.18
• Chapter 4: Exercises: 4.1, 4.2, 4.14-4.16, 4.18-4.21
• Chapter 5: Exercises: 5.1-5.4, 5.6-5.8, 5.11-5.13
• Chapter 6: Exercises: 6.1, 6.3-6.7, 6.15-6.17
• Chapter 7: Exercises: 7.1-7.12, 7.14-7.21