## MAT 311 Introduction to Number Theory, Spring 2009.

• Instructor: Olga Plamenevskaya, office 3-107 Math Tower, e-mail: olga@math.sunysb.edu
• Office hours: Monday 12:45am-1:45pm in P-143, Monday 1:45-3:45 in 3-107, or by appointment.
• Class meetings: MWF, 10:40-11:35pm, Physics 127.
• References :
I. Niven, H. Zuckerman, H. Montgomery,An Introduction to the Theory of Numbers, 5th edition.
This is the required text; parts of the homework will be assigned from it.

• Exams: there will be two midterm exams and a final exam (TBA).

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.

Important: Please 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. Some solutions
Homework 3 (pdf), due Feb 18. Some solutions
Homework 4 (pdf), due Feb 25. Solutions
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. Solutions
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! Solutions
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. Solutions
Homework 10, due Mar 25: 1, 2, 3 sec. 7.4; 3, 4 sec. 7.5; 1 sec. 7.6. Some solutions

• 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