01:640:356 Theory of Numbers (Rutgers University, Spring 2023)

Main goals: To understand fundamental facts about numbers on both intuitive and rigorous level. This would be useful when reading or doing mathematics of all kinds (e.g., algebra, combinatorics, calculus, logic). Some practical applications will be also discussed.

Time and Place: Monday and Thursday 12:10 PM - 1:30 PM; Livingston Campus, BE-201, Rutgers University.
Textbook: Kenneth H. Rosen; Elementary Number Theory and Its Applications (6th edition) + other sources;
Lecturer: Stoyan Dimitrov (emailТoStoyan {at} gmail [dot] com).
Office Hours: Friday, 3:40pm - 5pm via Zoom (write me an email beforehand) or by appointment.

Syllables:

Week	Topic	Sources
1	Integers and sequences || Sums, products and figurate numbers	§1.1-1.2 + this file
2	Mathematical Induction (weak and strong) || Fibonacci numbers	§1.3-1.4 + this file
3	Integer representations	§2
4	Prime numbers and Euclid's algorithm	§3
5	Modular arithmetic and the Chinese Remainder Theorem	§4
6	Divisibility tests, Willson's theorem and Fermat's little theorem	§5-6
7	Pseudoprimes and probabilistic primality tests	§6.2
8	Midterm
9	Euler's Phi-function, sum and number of divisors || Perfect numbers, Mersenne primes	§7.1-7.3
10	The Möbius function and The Möbius inversion formula	§7.4
11	Euler's Theorem and the RSA algorithm	§8
12	Quadratic residues and the law of Quadratic Reciprocity	§11
13	Integer Partitions I&II	§7.5
14	Integer Partitions III	this file + Bressoud-Zeilberger great proof
15	Continued fractions + review	§12 + this file + this exciting proof

Grading scheme:

[35%] Final exam, May 6 (Monday)
[25%] Midterm exam, March 9 (Thursday)
[20%] Homework 1 - click here (due date - week 9, the Thursday section (print out your solutions))
[20%] Homework 2 (due date - week 15, the Thursday section)

Additional comments:

• This is the official site of the course. The Canvas page will be used only for announcements and to enter grades.
• A makeup exam or quiz will only be given in the rare case of a student who has a documented emergency, and has made every reasonable effort to contact the professor before the exam.
• Plagiarism and cheating in any form is strongly prohibited! All students in the course are expected to be familiar with and abide by the academic integrity policy
• Rutgers is fully committed to compliance with the Americans with Disabilities Act; policies and procedures are indicated at http://ods.rutgers.edu. Students who wish to request special accommodations must present a Letter of Accommodations to the instructor as early in the terms as possible.
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