Discrete Mathematics

1. Number Theory
   1. Divisibility and Primes
      1. Divisibility Rules
         1. Definitions and Basic Properties
         2. Application of Divisibility Rules
      2. Greatest Common Divisor (GCD)
         1. Euclidean Algorithm
            1. Iterative Method
            2. Recursive Method
         2. Extended Euclidean Algorithm
            1. Finding Inverses
            2. Applications in Solving Linear Diophantine Equations
      3. Least Common Multiple (LCM)
         1. Relationship with GCD
         2. Calculating LCM using Prime Factorization
      4. Prime Numbers
         1. Definition and Properties
         2. Primality Tests
            1. Trial Division
            2. Fermat Primality Test
            3. Miller-Rabin Primality Test
         3. Distribution of Primes
            1. Prime Number Theorem
            2. Bertrand's Postulate
      5. Prime Factorization
         1. Uniqueness of Prime Factorization (Fundamental Theorem of Arithmetic)
         2. Applications of Prime Factorization
            1. Simplifying Fractions
            2. LCM and GCD Calculations
   2. Modular Arithmetic
      1. Congruences
         1. Basic Properties and Definitions
         2. Linear Congruences
            1. Methods for Solving Linear Congruences
            2. Applications in Cryptography and Coding Theory
      2. Theorems in Modular Arithmetic
         1. Fermat’s Little Theorem
            1. Statement and Proof
            2. Applications in Number Theory and Cryptography
         2. Wilson’s Theorem
            1. Statement and Applications
         3. Euler's Theorem
            1. Connection to Fermat's Little Theorem
            2. Proof and Applications
      3. Chinese Remainder Theorem
         1. Statement and Applications
         2. Solving Systems of Congruences
         3. Use in Computer Science and Cryptography
      4. Modular Inverses
         1. Computing Using Extended Euclidean Algorithm
         2. Applications in Solving Equations
   3. Number Theoretic Functions
      1. Euler's Totient Function
         1. Definition and Basic Properties
         2. Calculation Methods
         3. Using Totient Function in Cryptography (RSA Algorithm)
      2. Divisor Function
         1. Number of Divisors
         2. Applications and Properties
         3. Sum of Divisors
      3. Möbius Function
         1. Definition and Properties
         2. Möbius Inversion Formula
         3. Role in Number Theory
      4. Sigma Function
         1. Definition and Calculation
         2. Relationship with Divisor Function
      5. Perfect Numbers and Amicable Numbers
         1. Definitions and Examples
         2. Historical Context and Modern Research
   4. Diophantine Equations
      1. Linear Diophantine Equations
         1. General Solutions and Methods
         2. Applications in Cryptography
      2. Pythagorean Triples
         1. Generating Pythagorean Triples
         2. Relationship with Geometry
      3. Pell’s Equation
         1. Method of Solution
         2. Historical Significance
   5. Advanced Topics in Number Theory
      1. Quadratic Residues
         1. Legendre Symbol
         2. Quadratic Reciprocity
      2. Continued Fractions
         1. Representation of Numbers
         2. Applications and Convergents
      3. Algebraic Number Theory
         1. Rings of Integers
         2. Ideals and Factorization
         3. Applications in Solving Equations
      4. Analytic Number Theory
         1. Distribution of Primes
         2. Riemann Zeta Function
         3. Applications and Modern Research
      5. Additive Number Theory
         1. Sum of Two Squares Theorem
         2. Goldbach’s Conjecture
         3. Waring’s Problem