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 (due date - week 9, the Thursday section)
[20%] Homework 2 (due date - week 15, the Thursday section)

Additional comments: