I taught a Continuation of this course, NT2, in Spring 2001.
Authors: | Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery | ISBN: | 978-0-471-62546-9 |
Year: | 1991 | Publisher: | John Wiley & Sons |
Here is The Publisher's site and at Amazon.com.
This is a classic number theory textbook
(Niven & Zuckerman
),
updated by Hugh Montgomery. It is renowned for its excellent problems.
I hope to cover chapters 1-4, and parts of chap. 5. (If a group of students have interest in another part of the book, I am willing to juggle topics. Most likely, you should see during the first third of the semester.)
Prof. Hugh Montgomery is maintaining an errata sheet (pdf) for the text.
Prof. Ken Ribet's has a guide to
recent and classic books
on number theory. The references, on his webpage, to
“Math 115”
refer to his number theory course, not mine!
Note that I
and my
are Prof. Ribet speaking.
(2000) | |
---|---|
Mon. 07Feb | Exam-Z (pdf): [In class.] The exam may cover material through the end of the Binomial section, NZM 1.4. 16Feb2000: Here are Solns to Exam-Z (txt). |
Fri. 18Feb | Quiz Q1: Soln to Q1 (txt). |
Wed. 22Mar | Exam-Y (pdf): Held in room LIT368 from 5PM-6:30PM. Please bring a hand-held calculator and lots of paper. This is an open-brain closed-book exam. [We went over solutions in class.] |
TBS | Exam-X (pdf): There will be no final-exam during exam week. |
Welcome! This is a continuation of my Number Theory 1; it should be accessible to anyone with an introductory course or who has read the first few chapters of a beginning text (E.g. Strayer, Vanden Eynden).
Here are text files for the first exam; I also emailed out this material. There are 6 questions, Q0,Q1,…,Q5. q0-1.NT2001g (txt), q2-NT2001g (txt), q3-NT2001g (txt), and q4-5.NT2001g (txt).
I plan to run this more as a seminar than as a standard Learn/Exam/Learn/Exam… course. While I expect that I will present the lion's share of the material, I hope to encourage students to perhaps give some prepared talks on a particular problem or subject.
As such, I expect that every student who attends regularly and participates in the discussions will earn an A. There will be some graded homework, and you can do homework singly or jointly, as you choose.
One of the seminal mathematics results this century (ahem, which ends midnight of 31Dec2000) is Prof. Wiles proof of Fermat's Last Theorem.
Wiles proof uses elliptic curves as a main ingredient. While we will not be able to cover all the details of the proof, we will be able to cover its broad outline, and will learn fascinating facts -discovering a hidden group- concerning Elliptic curves.
Along the way we will solve a number of Diophantine equations, that is,
algebraic equations where the only solutions that we allow are using integers
(sometimes rational numbers are allowed). In particular
we will find all Pythagorean triples
(a,b,c) of positive integers for which
a^{2} + b^{2} = c^{2} .
We will discover that there is a two-parameter family of such solutions.
We will prove Lagrange's theorem that every positive integer is a sum of exactly 4 perfect squares (allowing the square of zero). Exercise: How is 150 a sum of four squares? Lagrange's proof using the beautiful and elementary "proof by Infinite Descent" method of Fermat. It gives an algorithm for finding four such squares.
We plan to cover much of Chapter 5 of our textbook An Introduction to the Theory of Numbers (5^{th} edition) by Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery.
A review of modular arithmetic and the Chinese Remainder Thm. The Legendre and Jacobi symbols. We'll then start solving some simple Diophantine equations (DE), roughly following chapter 5 of NZM.