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Elementary Number Theory II
Math 366 - Spring 2020


Announcements

For the rest of the semester we will be using odtuclass and gradescope. You should check your emails regularly for important announcements.

The information below may not be accurate since it wasn't updated after the Coronavirus break.


Exercise Sets

I will assign exercise questions on a regular basis. I strongly suggest that you shall work on these questions.

Exercise Set 1

Exercise Set 2

Exercise Set 3

Exercise Set 4


Exams And Quizes

There will be three exams and several quizes (seven to ten) throughout the semester.

Quiz 1

Quiz 2

Midterm 1


Office Hours

Monday 13:40-15:30


Grading

Your final letter grade will be determined by three exams and several quizes.

Midterm 1, 30 points - March 4 (during lecture hours)

Midterm 2, 30 points - April 1 (during lecture hours)

Final, 30 points - To be announced

Quizzes, 10 points - in lectures


Introduction

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to take integer values only. The following are traditional problems related with Diophantine equations:

1st --> Are there any solutions?

2nd --> If so, finitely or infinitely many?

3rd --> Can all solutions be found?

An example: Consider the Diophantine equation

x2   +   (x+1)2   =   y2.

The answer to the 1st question is affirmative since a quick computer search gives:

02+12=12

32+42=52

202+212=292

1192+1202=1692

6962+6972=9852

In order to answer the 2nd question, we need to construct an infinite family of solutions. Consider the recurrence relation

an = 6an-1 - an-2 + 2,    a0 = 0,     a1 = 3.

This is a famous sequence, see A001652. One can use induction to see that an2+(an+1)2 is always a square for each natural number n.

We still need to answer the 3rd question. It turns out that, we need to know the solutions of x2-2y2=1 for this. This last equation is an example of a Pell equation and its solutions are related with units of the ring Z[√2].

In this course we will investigate several Diophantine equations. We will go over some basic examples in the first half of the semester, then we will continue with the powerful methods of algebraic number theory in the second half.

If time permits, we will study the transcendental numbers at the end of the course and prove that e and π are transcendental.


Prerequisite

If you are planning to take this course, you should know

Arithmetic in Integers (taught in 116, 365): Divisibility, greatest common divisor, unique factorization, congruences, etc...

Algebraic Structures (taught in 116): Groups, rings, fields, etc...


References

The main reference for the course will be your lecture notes. I will consult the following books for preparing my notes. I may change their terminology and notation and cover some extra topics that are not available in these books.

W. W. Adams and L. J. Goldstein. Introduction to number theory.

J. H. Silverman and J. Tate. Rational Points on Elliptic Curves.

D. M. Burton Elementary number theory, 7th edition.

I. Stewart and D. Tall. Algebraic number theory and Fermat's last theorem, 3rd edition.


Tentative Course Outline

Diophantine equations:

Week 1 - Introduction.

Week 2 - Pythagorean triples (Adams&Goldstein 6.2 and Burton 12.1).

Week 3 - Elliptic curves (Silverman&Tate Chapters 1 and 2).

Week 4 - Fermat's infinite descent (Adams&Goldstein 6.3 and Burton 12.2).

Week 5 - Sums of squares (Adams&Goldstein 6.4, 6.5 and Burton 13.2, 13.3)

Week 6 - Pell's equation (Adams&Goldstein 6.6).

Week 7 - Continued fractions (Burton 15.2, 15.3, 15.5).

Algebraic number theory and quadratic fields:

Week 8 - The Gaussian integers (Adams&Goldstein 7.1, 7.2).

Week 9 - The Gaussian integers (Adams&Goldstein 7.2, 7.3).

Week 10 - Algebraic numbers and integers (Adams&Goldstein 8.2, 8.3)

Week 11 - Factorization into irreducibles (Stewart&Tall Chapter 4).

Week 12 - The arithmetic of ideals (Stewart&Tall Chapter 5).

Week 13 - Class group and class number (Stewart&Tall Chapters 9 and 10)

Transcendental numbers:

Week 14 - Algebraic and transcendental numbers. Transcendence of e and π.


Course Policy

Attendance will be taken and if your attendance is less than %70, you will not be able to take the final exam and you will get the NA grade. Medical reports less than five days will not be accepted.

Only one make-up examination will be offered for each student. The excuse for not attending an examination must be proved with documents. A make-up examination will take place a few days after the missed exam.

The quizes will be given in lectures and may not be announced in advance. A portion of quiz scores will be dropped and there will be no make-up for quizes.