Unit 3072: Comprehensive Number Theory and Applications

1. Prime Numbers

1.1 Definition and Fundamental Properties

• What are prime numbers?
• Unique factorization theorem

1.2 Identification of Prime Numbers

• Trial division
• Sieve of Eratosthenes
• Probabilistic tests (brief mention)

1.3 Prime Factorization

• Breaking down numbers into primes
• Applications in simplifying fractions

1.4 Importance in Number Theory and Cryptography

• Role in RSA encryption
• Prime distribution overview

2. Divisibility Rules

2.1 Basic Divisibility Rules

• Divisibility by 2, 3, 4, 5, 6, 9

2.2 Divisibility by Other Common Divisors

• Rules for 7, 8, 10, 11

2.3 Techniques and Applications

• Using rules to factor numbers efficiently
• Simplifying expressions

3. Modular Arithmetic

3.1 Introduction to Modulus and Congruence

• Definition of modular arithmetic
• Notation and properties of congruence

3.2 Modular Operations

• Addition, subtraction, multiplication modulo n
• Modular exponentiation techniques

3.3 Applications

• Cryptography basics (e.g., modular exponentiation in RSA)
• Computer science applications

4. Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

4.1 Definitions and Concepts

• Understanding GCD and LCM
• Relationship between GCD and LCM

4.2 Computing GCD

• Prime factorization method
• Euclidean algorithm

4.3 Computing LCM

• Using prime factorization
• Relation to GCD

4.4 Applications

• Simplifying fractions
• Solving linear Diophantine equations

5. Diophantine Equations

5.1 Introduction

• Definition and importance
• Types of polynomial Diophantine equations

5.2 Methods of Solution

• Factorization techniques
• Using modular arithmetic
• Introduction to Pell's equation

5.3 Examples and Problem Solving

• Linear Diophantine equations
• Quadratic forms

6. Fermat's Little Theorem and Euler's Totient Function

6.1 Fermat's Little Theorem

• Statement and proof sketch
• Applications in primality testing and cryptography

6.2 Euler's Totient Function

• Definition and calculation methods
• Properties and examples

6.3 Relationships and Applications

• Generalization of Fermat’s theorem
• Use in RSA algorithm

7. Pythagorean Triples

7.1 Definition and Basic Properties

• The Pythagorean theorem refresher
• What are Pythagorean triples?

7.2 Generating Pythagorean Triples

• Euclid's formula
• Primitive vs non-primitive triples

7.3 Classification and Applications

• Connections to geometry and number theory
• Examples and problem-solving

8. Number Systems

8.1 Overview of Number Systems

• Decimal, binary, octal, hexadecimal

8.2 Conversion Between Number Systems

• Techniques for conversion
• Examples

8.3 Arithmetic Operations in Different Bases

• Addition, subtraction, multiplication, division
• Use in computer science and digital electronics

8.4 Applications

• Data representation
• Programming and logic circuits