MAT 311 Introduction to Number Theory, Spring 2009.
- Instructor: Olga Plamenevskaya, office 3-107 Math Tower,
- Office hours: Monday 12:45am-1:45pm in P-143, Monday 1:45-3:45 in 3-107, or by
- Class meetings: MWF, 10:40-11:35pm, Physics 127.
This is the required
text; parts of the homework will be assigned from it.
- I. Niven, H. Zuckerman, H. Montgomery,An Introduction to the Theory of Numbers, 5th edition.
Exams: there will be two midterm exams and a final exam
Midterm I: Monday, Mar 2, in class.
Midterm II: Monday, Apr 20, in class.
The second exam will cover the material learned since the first exam: polynomial congruences,
primitive roots and power residues, quadratic residues and reciprocity, Legendre/Jacobi symbols,
sums of squares, Pythagorean triples, and some ad hoc methods of solving equations in integers (sec. 5.4.)
Homework: weekly assignments will be posted on
this page. Homework will constitute a significant part of your course grade.
write up your solutions neatly, be sure to put your name on the first page and staple all pages.
Illegible homework will not be graded.
You are welcome to collaborate with others and to consult books,
but your solutions should be written up in your own words,
and all your
collaborators and sources should be listed.
Homework 1 (pdf), due Feb 4. Some solutions
Homework 2 (pdf), due Feb 11.
Homework 3 (pdf), due Feb 18.
Homework 4 (pdf), due Feb 25.
Homework 5 (pdf), due Mar 4.
Homework 6, due Mar 18: 4,7 sec. 2.6; 3 sec. 2.7; 2, 9 sec. 2.8.
Homework 7, due Mar 25: 8, 10, 18, 20, 22, 37 sec. 2.8.
Hint for question 37: consider two cases, n even and n odd. For n odd,
let p be the least prime divisor of n, and show that (p-1,n)=1. Then think
about the order of 2 modulo p. Solutions
Homework 8, due Apr 15: 2, 9, 10 sec. 3.2; 1, 2 sec. 3.3; 8 sec. 5.3,
and one more question: find all integers that can be represented as a difference of two squares, i.e.
n=a2-b2 . This is a lot easier than sums of squares!
Homework 9 (pdf), due Apr 29. There's a typo on
the problem sheet, I meant to assign qiestion 3 from section 7.3 not 7.2.
Homework 10, due Mar 25: 1, 2, 3 sec. 7.4; 3, 4 sec. 7.5; 1 sec. 7.6.
Prerequisites: I will try to keep prerequisites to a minimum, but
some familiarity with abstract algebra and number theory will be assumed.
Syllabus: we will cover a number of topics from elementary number theory
(selected chapters of the texbook). Time permitting, the topics will include
- Divisibility, Euclidean algorithm,
- Prime numbers, Fundamental theorem of arithmetic
- Linear Diophantine equations / Congruences
- Fermat's Little Theorem, Euler's Formula
- Powers modulo m
- Public key cryptography
- Primitive roots
- Quadratic residues
- Sums of squares
- Farey fractions, rational approximation, continued fractions
- Elliptic curves
Students with Disabilities: If you have a physical,
psychological, medical, or learning disability that may impact on your
ability to carry out assigned course work, you are strongly urged to
contact the staff in the Disabled Student Services (DSS) office: Room
133 in the Humanities Building; 632-6748v/TDD. The DSS office will
review your concerns and determine, with you, what accommodations are
necessary and appropriate. A written DSS recommendation should be
brought to your lecturer who will make a decision on what special
arrangements will be made. All information and documentation of
disability is confidential. Arrangements should be made early in the
semester so that your needs can be accommodated.