Here’s a list of 100 chapter titles for Number Theory Algorithms tailored for competitive programming, organized from beginner to advanced levels:
- Introduction to Number Theory in Competitive Programming
- Divisibility and Basic Properties
- Prime Numbers: Definition and Identification
- Sieve of Eratosthenes: Finding Primes Efficiently
- Prime Factorization: Breaking Down Numbers
- Greatest Common Divisor (GCD): Euclidean Algorithm
- Extended Euclidean Algorithm: Solving Linear Diophantine Equations
- Least Common Multiple (LCM): Properties and Applications
- Modular Arithmetic: Basics and Properties
- Modular Addition, Subtraction, and Multiplication
- Modular Exponentiation: Fast Powering
- Multiplicative Inverses: Definition and Applications
- Fermat's Little Theorem: Introduction and Proof
- Euler's Totient Function: Definition and Properties
- Euler's Theorem: Generalization of Fermat's Little Theorem
- Chinese Remainder Theorem: Introduction and Applications
- Basic Problems on Prime Numbers
- Basic Problems on GCD and LCM
- Basic Problems on Modular Arithmetic
- Introduction to Number Theory in Problem Solving
- Handling Large Numbers: Big Integer Operations
- Sum of Divisors Function: Basics
- Number of Divisors Function: Basics
- Perfect Numbers: Definition and Examples
- Amicable Numbers: Introduction
- Pythagorean Triples: Generating and Verifying
- Fibonacci Numbers: Properties and Applications
- Catalan Numbers: Introduction
- Factorials and Their Properties
- Basic Number Theory Problems in Competitive Programming
- Advanced Sieve Techniques: Segmented Sieve
- Pollard's Rho Algorithm: Factoring Large Numbers
- Miller-Rabin Primality Test: Probabilistic Primality Checking
- Lucas-Lehmer Test: Primality of Mersenne Primes
- Carmichael Numbers: Pseudoprimes
- Wilson's Theorem: Applications in Primality Testing
- Legendre's Formula: Prime Factorization of Factorials
- Mobius Function: Definition and Applications
- Inclusion-Exclusion Principle in Number Theory
- Sum of Divisors Function: Advanced Techniques
- Number of Divisors Function: Advanced Techniques
- Linear Diophantine Equations: Advanced Problems
- Chinese Remainder Theorem: Advanced Applications
- Discrete Logarithms: Baby-Step Giant-Step Algorithm
- Primitive Roots: Definition and Properties
- Quadratic Residues: Introduction
- Legendre Symbol: Definition and Applications
- Jacobi Symbol: Generalization of Legendre Symbol
- Tonelli-Shanks Algorithm: Solving Quadratic Congruences
- Continued Fractions: Basics and Applications
- Pell's Equation: Solving and Applications
- Sum of Squares: Representing Numbers
- Sum of Cubes: Representing Numbers
- Advanced Problems on GCD and LCM
- Advanced Problems on Modular Arithmetic
- Advanced Problems on Prime Numbers
- Advanced Problems on Euler's Totient Function
- Advanced Problems on Multiplicative Inverses
- Advanced Problems on Chinese Remainder Theorem
- Intermediate Number Theory Problems in Competitive Programming
- Sieve of Atkin: Advanced Prime Generation
- Quadratic Sieve: Factoring Large Numbers
- Number Field Sieve: Factoring Very Large Numbers
- AKS Primality Test: Deterministic Primality Checking
- Elliptic Curve Primality Proving
- Advanced Applications of Euler's Totient Function
- Advanced Applications of Mobius Function
- Advanced Applications of Inclusion-Exclusion Principle
- Advanced Applications of Chinese Remainder Theorem
- Advanced Applications of Discrete Logarithms
- Advanced Applications of Primitive Roots
- Advanced Applications of Quadratic Residues
- Advanced Applications of Legendre and Jacobi Symbols
- Advanced Applications of Tonelli-Shanks Algorithm
- Advanced Applications of Continued Fractions
- Advanced Applications of Pell's Equation
- Advanced Applications of Sum of Squares and Cubes
- Advanced Problems on Linear Diophantine Equations
- Advanced Problems on Modular Exponentiation
- Advanced Problems on Multiplicative Functions
- Advanced Problems on Prime Factorization
- Advanced Problems on Sieve Techniques
- Advanced Problems on Primality Testing
- Advanced Problems on Factorials and Divisors
- Advanced Problems on Fibonacci and Catalan Numbers
- Advanced Problems on Pythagorean Triples
- Advanced Problems on Perfect and Amicable Numbers
- Advanced Problems on Carmichael Numbers
- Advanced Problems on Wilson's Theorem
- Advanced Number Theory Problems in Competitive Programming
- Advanced Topics in Elliptic Curves
- Advanced Topics in Algebraic Number Theory
- Advanced Topics in Analytic Number Theory
- Advanced Topics in Computational Number Theory
- Advanced Topics in Cryptography and Number Theory
- Advanced Topics in Diophantine Approximation
- Advanced Topics in Modular Forms
- Advanced Topics in L-Functions
- Advanced Topics in Zeta Functions
- Open Problems in Number Theory and Competitive Programming
This structured progression ensures a comprehensive understanding of number theory algorithms, from foundational concepts to cutting-edge techniques, all tailored for competitive programming.