### MAT 311 Number Theory

Fall 2021

We will meet on MW 2:40pm-4:00pm in Mathematics 4130

First day of class: Monday August 23, 2021.
Final exam : Final Exam: Wednesday, Dec. 8, 5:30pm-8:00pm

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 10:00am-11:00am, W 9:00am-10:00am in MLC; M 10:00am-11:00am in his office Math Tower S-235M

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 two-dimensional 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

 Exam I 25% Exam II 25% Final Exam 35% Homework 15%

Late homework will not be accepted.

Student Accessibility Support Center Statement:

If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, Stony Brook Union Suite 107, (631) 632-6748, or at sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the Student Accessibility Support Center. For procedures and information go to the following website: https://ehs.stonybrook.edu//programs/fire-safety/emergency-evacuation/evacuation-guide-disabilities and search Fire Safety and Evacuation and Disabilities.