Michael MOVSHEV 
MAT 311

We will meet on MW 2:40pm4:00pm in Mathematics 4130
First day of class: Monday August 23, 2021.
Final
exam : Final Exam: Wednesday, Dec. 8, 5:30pm8:00pm
Office hours:
TBA.
How to contact me?
the best way is to email me there:
mmovshev at math
dot sunysb dot edu
The grader Daniel Brogan has office hours:F 10:00am11:00am, W 9:00am10:00am in MLC; M 10:00am11:00am in his office Math Tower S235M
Our textbook:
An
Introduction to the Theory of Numbers (Hardcover), Wiley, Fifth
edition (January 1991), by Ivan Niven, Herbert S. Zuckerman, Hugh L.
Montgomery
Link to Current Homework: The Homework is an important part of this class. I will take it from the book or from other sources. Click here to go to the homework page.
Course notes and announcements:
Quick intro: Number theory is certainly one of the oldest
subject within mathematics. Already 36 centuries ago in tablets
written in Babylone, there were examples of such problems. Some
mathematicians like to say that it occupies within mathematics the
same place as mathematics within science...Some people like to see it
as the purest domain in mathematics, and yet some others like to see
all its applications to cryptography, computer science,etc...
Number
theory has the remarkable advantage of being able to formulate
extremely deep problems almost without prerequisites. A model for
this is certainly Fermat's last theorem, that can be stated in one
line but that resisted all the efforts of mathematicians for
centuries... For this reason, I think it is an excellent "entry
point" to mathematics: we will start with very simple material
like divisibility properties, congruences, continue with simple
Diophantine equations, and slowly progress towards deeper questions
like Quadratic reciprocity.
I will not hesitate to provide
introductions to much recent material, like one and twodimensional
representations, or even the Absolute Galois group, which is nowadays
one of the most mysterious objects of contemporary mathematics, and
one that is certainly the center of a tremendous mathematical
activity.
Prerequisites:
For this class you need to have taken MAT
312 or 313 or 318.
Link to Current Homework: Regularly you will have to consult this homework page to know what has been assigned.
Syllabus :
Day of 
Sections Covered 
Unit 1 
Divisibility and Congruences 
Week 1:Aug. 23 
Section 1.1 Introduction, Section 1.2 Divisibility 
Week 1:Aug. 25 
Section 1.3 Primes, Section 1.4 The Binomial Theorem 
Week 2:Aug. 30 
Section 2.1 Congruences, Section 2.2 Solutions of Congruences 
Week 2: Sept 1 
Section 2.3 The Chinese Remainder Theorem 
Week 3:Sept. 6 
Labor Day Classes not in Session 
Week 3:Sept. 8 
Section 2.6 Prime Power Moduli 
Week 4:Sept. 13 
Section 2.7 Prime Modulus 
Week 4:Sept. 15 
Section 2.8 Primitive Roots and Power Residues 
Week 5:Sept. 20 
Section 3.1 Quadratic Residues 
Unit 2 
Quadratic Reciprocity and Diophantine Equations 
Week 5:Sept. 22 
Section 3.2 Quadratic Reciprocity 
Week 6:Sept. 27 
Section 5.1 Binary Linear Forms, Section 5.2 Simultaneous Linear Equations 
Week 6:Sept. 29 
Midterm 1 on Unit 1 
Week 7:Oct 4 
Section 3.4 Binary Quadratic Forms 
Week 7:Oct 6 
Section 5.3 Pythagorean Triples 
Week 8:Oct 11 
Oct 11 Fall Break 
Week 8:Oct 13 
Section 3.5 Equivalence and Reduction of Binary Quadratic Forms 
Week 9:Oct 18 
Section 3.6 Sums of Two Squares 
Week 9:Oct 20 
Section 5.5 Ternary Quadratic Forms 
Week 10:Oct 25 
Midterm 2 on Unit 2 
Unit 3 
Special Cases of Fermat's Last Theorem 
Week 10:Oct 27 
Section 9.1 Polynomials, Section 9.2 Algebraic Numbers 
Week 11:Nov 1 
Section 9.3 Algebraic Number Fields, Section 9.4 Algebraic Integers 
Week 11:Nov 3 
Section 9.5 Quadratic Fields, Section 9.6 Units in Quadratic Fields 
Week 12:Nov 8 
Section 9.7 Primes in Quadratic Fields, Section 9.8 Unique Factorization, Section 9.9 Primes in Quadratic Fields with the UFD property 
Week 12:Nov 10 
Section 9.10 The Cubic Case of Fermat's Last Theorem 
Week 13:Nov 15 
Selected Topics from Chapter 7 
Week 13:Nov 17 
Selected Topics from Chapter 7 
Week 14:Nov 22 
Nov 24 Thanksgiving Break 
Week 14:Nov 24 
Selected Topics from Chapter 7 
Week 15:Nov 29 
Miscellaneous 
Week 15:Dec 1 
Miscellaneous 
Week 16:Dec 6 
Miscellaneous 
Exams:
Midterm I 
09/29/21 
Usual room 
Midterm II 
10/25/21 
Usual room 
Final 
12/08/21 
TBA 
Homework and grading policy: Here is how your final grade will be computed. of the following:
Exam I 
25% 
Exam II 
25% 
Final Exam 
35% 
Homework 
15% 
Late homework will not be accepted.
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