Elementary Number Theory II
Math 366 - Spring 2017
Announcements
Your final exam score is available in odtuclass. You can see your final exam paper on Tuesday and Wednesday (June 13-14).
Exercise Sets
Exams And Quizes
Office Hours
Tuesday 10:40-12:30
Thursday 12:40-13:30
Grading
Your final letter grade will be determined by three exams and several quizzes. The quizzes will be given in lectures and may not be announced in advance. The worst quiz score will be dropped and there will be no make-up for quizzes.
Midterm 1 (% 30 - March 30)
Midterm 2 (% 30 - April 27)
Final (% 30 - June 5)
Quizzes (% 10 - 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, 367, 365): Divisibility, greatest common divisor, unique factorization, congruences, etc...
Algebraic Structures (taught in 116, 367): Groups, rings, modules, 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.
I. Stewart and D. Tall. Algebraic number theory and Fermat's last theorem.
Course Policy
Attendance will be taken and if your attendance is less than %60, you will not be able to take the final exam and you will get the NA grade.
Only one make-up examination will be offered. The excuse for not attending an examination must be proved with documents. The make-up examination will take place shortly after the final exam.
Tentative Course Outline
A few Diophantine equations:
Linear equations.
(Burton 2.4)
Pythagorean triples.
(Adams 6.2 and Burton 11.1)
Elliptic curves.
(Silverman Chapters 1 and 2)
Fermat's infinite descent.
(Adams 6.3 and Burton 11.2)
Representation of integers as sums of squares.
(Adams 6.4, 6.5 and Burton 12.2, 12.3)
Pell's equation, continued fractions.
(Adams 6.6 and Burton 14.2, 14.3, 14.4)
Algebraic number theory and quadratic fields:
The Gaussian integers.
(Adams 7.1, 7.2, 7.3)
Algebraic numbers and integers.
(Adams 8.2, 8.3)
Factorization into irreducibles. Bachet and Ramanujan-Nagell equations.
(Stewart Chapter 4)
The arithmetic of ideals. Class group and class number.
(Stewart Chapters 5, 9 and 10)
Primes of the form p = x^2 + ny^2. Fermat's last theorem.
(A very brief summary)
Transcendental numbers: Algebraic and transcendental numbers. Transcendence of e and π.