# Math 25 (Fall 2010): Number Theory

## News

• 12/1/2010: The room for the final exam is Carson L02. You can also find this information on the registrar's website.
• 11/5/2010: The content of the second midterm will be chapters 3, 4, and 5 of the textbook.
• 10/8/2010: The content of the first midterm will be everything in chapters 1 and 2 of the textbook, and also induction. Also, there's a section of the webpage dedicated to Internet sites about things we've talked about in class. Visiting these sites is purely optional, but you'll find lots of interesting information!
• 9/29/2010: The midterms will indeed be on the dates indicated below, in Moore Hall B03.
• 9/27/2010: Preliminary times for the two midterms are as follows: Thursday, October 14, 2010, 6pm - 8pm, and Thursday, November 11, 6pm - 8pm. Location will be announced once it becomes available.

Class Notes, Homework assignments, external links to various topics briefly discussed in class.

## Introduction

Math 25 is an introduction to the subject of number theory. It is not too much of a simplification to say that number theory is the mathematical study of integers and rational numbers, as opposed to the study of real numbers (calculus) or complex numbers. Number theory is a vast and ancient subject, and it is no exaggeration to say that one could spend a lifetime studying it and still not come close to understanding the entire field, but the formal requirements for beginning the study of number theory are minimal. We will survey a variety of elementary but important results, and hopefully illustrate some applications that even basic number theory has in real life.

The ORC prerequisite for this class is Math 8 (Calculus II), but in reality virtually no calculus is needed in this class. Perhaps the most important determinant of success in a class such as this is the willingness to think deeply and diligently about mathematics - in other words, hard work! Students who have seen some algebra (Math 31, 71, 81) will recognize a nontrivial intersection between algebra and number theory, but we do not assume any knowledge of algebra in this class. We may point out some places where algebraic ideas naturally appear, so in some sense this class might serve as motivation for ideas from algebra.

Besides teaching you about number theory, a secondary objective in this class is to teach you how to write clear and logical solutions to mathematical problems. Some homework assignments may ask you to compute a certain number (akin to problems in calculus or linear algebra, where you may calculate integrals or solve systems of linear equations). In these problems, not only should you find the correct number, you should also explain how you arrived at your answer. Other questions may ask you to show that a certain statement is either true or false. In these problems, you need to give clear, logically correct explanations - in other words, proofs. A substantial part of the lectures in this class will be devoted to giving detailed proofs of mathematical statements.

Name: Andrew Yang
Office: Kemeny 316
Office Hours: Monday, Wednesday, Friday 2:30pm - 3:30pm, or by appointment

The grader for this class is Hee-Sung Yang, '12.

## Examination Schedule

The exams will be during the following dates:

• Thursday, October 14, 2010, 6pm - 8pm, Moore Hall B03
• Thursday, November 11, 2010, 6pm - 8pm, Moore Hall B03
• Sunday, December 5, 2010, 3pm - 6pm, Carson L02

If you are unable to be at any of these exams, please contact me as soon as possible so we can setup alternate test-taking arrangements.

Your grade in this class will be determined by homework and exams. There will be weekly written homework assignments which will be turned in, graded, and returned to you. You should pay special attention to comments about the clarity and correctness of your written explanations.

There will also be two midterm exams and a final examination. The dates and times of these exams will be announced in the future. These exams will be closed book exams which will be of similar flavor to problems you are assigned for homework.

More precisely, homework will be 40% of your grade, the two midterms 30% (15% each), and the final 30%.

## Homework Policy

Homeworks will be posted on this website and will be usually due about a week after they are posted. Late assignments will only be accepted when granted an extension, which must be requested from the instructor several days in advance. In general, extensions will only be granted for health-related reasons or family emergencies. Exceptions may be made for school-related travel.

The homework collaboration policy for this class is more or less in line with other Dartmouth math classes. You are allowed to collaborate with others on homework, but must write your own solutions. A good rule of thumb is that you should never be copying phrases or sentences from anyone else or any source. You may use theorems, lemmas, etc. that we have covered from the textbook, but in general you should not use theorems, lemmas, etc. from sections of the book we have not covered or from external sources. Also, please write down the people you collaborated with and outside sources (namely, anything besides the required textbook) you consulted on your homework assignments.

## Textbook

The required book for this class is Elementary Number Theory, by Gareth A. Jones and J. Mary Jones, ISBN 3540761977.

There are many, many books written about elementary number theory, and even more books written about number theory in general. The following list of supplemental books is by no means all-inclusive:

• An Introduction to the Theory of Numbers, by G.H. Hardy and E.M. Wright. This is a classic introduction to number theory and is a great book to read for anyone serious about studying the subject. A distinguishing feature of this book is the complete lack of exercises.
• Elementary Number Theory, by Kenneth Rosen. This book has been used in some previous iterations of this class. It is long and has lots of exercises.
• An Introduction to the Theory of Numbers, by Ivan Niven, Herbert Zuckerman, and Hugh Montgomery. This book is also long and has lots of exercises, and is well-written.

## Course description

Our primary goal in this class is to cover through chapter 8 of the textbook. We begin by discussing divisibility, which naturally leads to the topic of prime numbers. This then leads to the study of "congruences modulo n", where n is a positive integer. In particular, we will consider analogues of familiar questions in algebra, such as solving linear and quadratic polynomials of a single variable, except instead of asking for solutions over real or complex numbers we will ask for solutions in some set of number-theoretic interest.

## Weekly schedule

This schedule is preliminary and will almost certainly be adjusted over the course of the term.

Week 1: Introduction

• An introduction to the subject of number theory.
• Mathematical notation, proof by induction and contradiction.
• Divisibility: basic properties.
• Reading: Beginning of Chapter 1

Week 2: Divisibility, prime numbers

• Divisibility: greatest common divisor and least common multiple, Euclid's algorithm.
• Solving linear equations in integers.
• Definition of prime numbers and basic properties, the fundamental theorem of arithmetic
• Reading: Chapter 1, beginning of Chapter 2

Week 3: Prime numbers, congruences

• Basic results on prime numbers
• Special classes of prime numbers (Fermat and Mersenne numbers)
• Congruences: definition, basic properties
• Reading: Chapter 2, beginning of Chapter 3

Week 4: Congruences continued

• Linear congruences, the Chinese remainder theorem
• Congruences modulo a prime
• Congruences modulo prime powers

Week 5: Congruences modulo prime powers

• Fermat's Little Theorem
• Pseudoprimes, Carmichael numbers, and a little about primality testing
• Lifting solutions, Hensel's Lemma

Week 6: The Euler totient function

• Definition
• Fermat-Euler Theorem
• RSA, basic cryptography

Week 7: The unit group

• Definition
• Primitive roots
• Classifying unit groups

• Definition, basic properties, Legendre symbol