Number Theory

Number theory is a branch of pure mathematics dedicated to studying the properties and relationships of integers, often called "the queen of mathematics" for its foundational elegance. It investigates core concepts like prime numbers, divisibility, and factorization, as well as more advanced topics such as modular arithmetic, Diophantine equations (polynomial equations seeking integer solutions), and the distribution of primes. While its origins are ancient, number theory has critical modern applications, most notably forming the bedrock of public-key cryptography systems that secure digital communication.

1. Foundations of Number Theory
   1. Historical Development of Number Theory
      1. Ancient Number Theory
      2. Medieval Contributions
      3. Modern Era Developments
      4. Contemporary Number Theory
   2. The Natural Numbers
      1. Definition and Properties
      2. Peano Axioms
      3. Mathematical Induction on Natural Numbers
   3. The Integers
      1. Definition and Construction
      2. Notation and Representation
      3. Algebraic Structure of Integers
         1. Closure Properties
         2. Associativity
         3. Commutativity
         4. Identity Elements
         5. Additive Inverses
         6. Distributive Property
      4. Ordering of Integers
         1. Order Relations
         2. Trichotomy Property
         3. Order Preservation
      5. Well-Ordering Principle
         1. Statement and Significance
         2. Applications in Proofs
         3. Relationship to Mathematical Induction
      6. Mathematical Induction
         1. Weak Induction
            1. Basis Step
            2. Inductive Step
            3. Proof Structure
         2. Strong Induction
            1. Definition and Method
            2. Comparison with Weak Induction
         3. Examples and Applications
            1. Algebraic Identities
            2. Divisibility Proofs
            3. Inequality Proofs