Math 105

Topics in Number Theory

Instructor: Carl Pomerance (carl.pomerance at dartmouth.edu)

### News and current assignment

Homework due Nov. 6: here.

Homework due Oct. 25: see this. (Note, a small edit was added to this at 2:30 pm on Oct. 23.)

Some notes of Florian Luca:
Diophantine Equations

Some notes of Michael Filaseta:
Transcendental number theory
(Parts of these course notes are fleshed out versions of arguments in Alan Baker, Transcendental Number Theory, Cambridge U. Press, 1975.)

Check out the article "Solving the Pell equation" by H. W. Lenstra, jr, in Notices Amer. Math. Soc. 49 (2002), 182-192.

We will start out using the text
Lecture Notes on Diophantine Analysis, by Umberto Zannier (Publications of the Scuola Normale Superiore / Lecture Notes (Scuola Normale Superiore)) [Paperback].
See a short list of errata here.
This book has not been ordered locally; it can be purchased at amazon.com and other online sellers. It can be bought fairly cheaply as far as textbooks go.

We meet Tuesday, Thursday 10:00 AM to 11:50 AM in 108 Kemeny.

Assignment 3, due Tuesday, Oct. 16:
From the book, exercises 2.22 - 2.27.

Assignment 2, due Thursday, Oct. 4:
From the book, excercises 2.1, 2.4, 2.5, and 2.6.

Assignment 1, due Thursday, Sept. 20.
From the book, do exercises 1.1, 1.2, 1.4, 1.5, and two of 1.7 to 1.10 (your choice).
Extra problems:
(1) Suppose m, n are integers that are not squares. Show that m1/2+n1/2 is irrational.
(2) Show that 21/2+31/3 is irrational. Generalize this as far as you can.

### Abstract

Introduction to Diophantine equations and Diophantine approximation:

The subject of Diophantine equations involves determining if a polynomial equation in several variables has any integer or rational solutions, and if it does, how to find them. Diophantine approximation refers to the approximation of real numbers with rational numbers. We shall see that the two topics are closely intertwined. Some of the more specific topics will be the Pell equation (quadratic in two variables) and the Thue equation (a homogeneous polynomial in two variables equal to a constant). We shall see how to prove that e and π are transcendental numbers, and we'll discuss some other pretty constants.

We will start out using the book "Lecture notes on Diophantine analysis" by Umberto Zannier. This book is available from amazon.com for about \$30 including shipping, but I'm not sure how robust the supply is. It would be good to order a copy soon. Partway into the term we'll branch out using some other material, which I'll make available.

There will be some written homework assignments, and students will be expected to occasionally present topics to the class, possibly a homework solution or a new topic. There will be no formal examinations. As far as prerequisites go, it would be good for students taking this class to know elementary number theory and some abstract algebra. Grading will be based on written assignments and in-class presentations (though grad students admitted to candidacy will not have their grades based on written assignments).

### Classes

Room: 108 Kemeny
Lectures: Tuesday-Thursday 10:00 am--11:50 am (10A hour)
X-hour: Wednesday 3:00 pm - 3:50pm

We may meet several of the x-hours, but this will always be announced in advance.

### Staff

Instructor:
Carl Pomerance -- 339 Kemeny / Tel. 6-2635
Office hours: None are regularly scheduled, but I'd be happy to make an appointment.
I have Math 8 hours MWF from 9 AM to 10 AM, and you're welcome to drop by then, but I'll give priority to Math 8 students.

### Textbook

Lecture Notes on Diophantine Analysis, by Umberto Zannier (Publications of the Scuola Normale Superiore / Lecture Notes (Scuola Normale Superiore)) [Paperback].

### Homework

Homework is due at the start of the class period on the due date.
Homework will be generally due once per week.
Assignments will be posted on this website, with extra problems and/or comments added as the week progresses.

### Exams

There will be no exams, but students will be expected to make some presentations to the class.