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Spr2000: MAS4203 3173 Intro.Number.Theory MWF5 LIT217

Number Theory 1 We may use more advanced computing devices...

I taught a Continuation of this course, NT2, in Spring 2001.

Handouts

A few purported solutions to NZM are available, namely, the infamous Problem P.191:25, [02Sep2015: Not quite; it is commented-out in the source file, as I need to work on the LaTeX.] which uses multiplicative-function ideas, as well as the Chinese RemainderThm.
The following problems,
   Problems P.17: 47,48,50, P.191: 23,24,25 and E-11 (pdf), are now available.
Our textbook is An Introduction to the Theory of Numbers (Fifth Edition).
Authors: Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery ISBN: 978-0-471-62546-9
Year: 1991 Publisher: John Wiley & Sons
Photo of text cover

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.

Syllabus

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.

[Image: Coffee cup] Homework and reading, so far.

Here are the E-problems (txt) (text file). You may present an E-prob whenever you wish: Simply come to class early and put your solution on the blackboard.

Homework assignments (txt).

Homework due Wedn., 20Mar2000 (pdf).

[Image: Steaming coffee cup] Exams and Projects (Number Thy)

There will be N exams, for some small positive integral value of N.
When(2000) ExamText
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.



Spr2001: MAT4930 5350 Number Theory2 (Special Topics) MWF3 LIT217

Number Theory 2 We may use more advanced computing devices...

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).

Exam A

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).

Exam B

Here is the Exam B (pdf, 2 pages), our final exam. For your reading pleasure you may scrutinize Purported Solns to Exam B (pdf).

Grading

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.

Our ambitious goal

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

a2 + b2  =  c2 .

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.

Syllabus

We plan to cover much of Chapter 5 of our textbook An Introduction to the Theory of Numbers (5th 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.

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____End: Number Theory