MATH 2409: Number Theory

Course Description

This course covers the fundamental principles and methods of Number Theory, one of the oldest and most attractive fields of mathematics. Topics covered include divisibility, primes, congruences, number-theoretic functions, quadratic residues, Diophantine equations, Fibonacci numbers, and continuing fractions. Applications in cryptography and current computational number theory are also investigated. Students will learn logical thinking, proof-writing, and problem-solving abilities, as well as how number theory applies to real-world applications like as encryption and computing.

Course Objectives

By the end of this course, students will be able to:

1. Apply divisibility, prime numbers, and the Euclidean method.
2. Solve congruences and use modular arithmetic to solve problems.
3. Demonstrate understanding of Fermat’s Little Theorem, Euler’s Theorem, and related concepts.
4. Use quadratic reciprocity and Legendre symbols to solve quadratic congruences.
5. Describe Diophantine equations, Pythagorean triples, and integer representations.
6. Use number-theoretic functions such as φ(n), τ(n), and σ(n).
7. Discover Fibonacci numbers, continuing fractions, and Pell’s equation.
8. Use number theory ideas for cryptography and primality testing.
9. Demonstrate logical reasoning and provide clear mathematical proofs for integers.

Required Materials

• • Primary Textbook: David M. Burton, Elementary Number Theory, 7th Edition, McGraw-Hill, 2011.
  • Niven, Zuckerman, & Montgomery – An Introduction to the Theory of Numbers
  • Rosen – Elementary Number Theory and Its Applications
  • Introduction to the Theory of Numbers, 1960 by G.H.; Wright, E.M. Hardy Publisher: Oxford University Press; Fourth Edition edition (1960) Language: English ISBN-10: 0198533101 ISBN-13: 978- 0198533108
  • Elementary Number Theory (Springer Undergraduate Mathematics Series) Corrected Edition by Gareth A. Jones; Publisher: Springer; Corrected edition (July 31, 1998) Language: English ISBN-10: 3540761977 ISBN-13: 978-3540761976