Here’s a list of 100 chapter titles covering Number Theory Basics in competitive programming, structured from beginner to advanced levels. These chapters will guide you through the fundamental concepts, algorithms, and advanced techniques of number theory, helping you solve various problems efficiently in competitive programming.
- Introduction to Number Theory in Competitive Programming
- Basic Definitions: Integers, Prime Numbers, and Divisibility
- Prime Numbers: What Are They and Why Are They Important?
- The Fundamental Theorem of Arithmetic
- Divisibility Rules: Quick Methods for Simple Checks
- Greatest Common Divisor (GCD): Definition and Properties
- Euclidean Algorithm for GCD
- Extended Euclidean Algorithm: Solving Linear Diophantine Equations
- Least Common Multiple (LCM): Definition and Calculation
- Properties of GCD and LCM
- Sieve of Eratosthenes for Finding Prime Numbers
- Prime Factorization of a Number
- Factorization Algorithms and Their Applications
- Understanding Modular Arithmetic
- Basic Modular Operations: Addition, Subtraction, Multiplication
- Modular Inverse: Definition and Computation
- Fermat's Little Theorem: Application in Modular Arithmetic
- Finding Modular Exponentiation Efficiently
- Chinese Remainder Theorem: Solving Systems of Modular Equations
- Basic Applications of Number Theory in Competitive Programming
- Divisors of a Number: How to Calculate Them
- Prime Factorization and Its Importance in Solving Problems
- Euler’s Totient Function: Definition and Calculation
- Euler’s Theorem and Its Application
- Modular Exponentiation: Efficient Algorithms for Large Numbers
- Fast Exponentiation Techniques: Binary Exponentiation
- Wilson’s Theorem: Testing Primes Using Factorials
- Perfect Numbers and Amicable Numbers
- Understanding Pairs of Numbers: Coprime and Relatively Prime
- Solving Problems Using GCD and LCM
- Efficient Calculation of GCD Using Binary GCD Algorithm
- Application of GCD in Linear Diophantine Equations
- Fermat's Little Theorem in Action: Fast Primality Testing
- Solving Problems Using Modular Arithmetic
- Chinese Remainder Theorem: Extended Algorithms and Applications
- Computing the Modular Inverse Using the Extended Euclidean Algorithm
- Finding Powers of Numbers Under Modulo Operation
- Greatest Common Divisor in Multiple Numbers
- Efficient Ways to Compute Large Powers in Modular Arithmetic
- Advanced Modular Arithmetic and Applications in Algorithms
- Advanced Modular Inverses and Their Applications
- Prime Testing Algorithms: Miller-Rabin and AKS Primality Test
- Sieve of Eratosthenes: Optimizing for Large Ranges
- Primality Testing: Efficient Approaches for Large Numbers
- Chinese Remainder Theorem for Non-Coprime Moduli
- Elliptic Curves and Their Applications in Cryptography
- Modular Arithmetic in Cryptographic Algorithms
- Number Theoretic Transform (NTT): Introduction and Applications
- Solving Quadratic Residues and Non-Residues
- Using Legendre Symbol to Simplify Modular Arithmetic
- Jacobi Symbol and Its Properties
- Linear Congruences: Solving Systems with Modular Arithmetic
- Quadratic Congruences and Solutions
- Solving Polynomial Congruences Efficiently
- Finding Square Roots Modulo Prime Numbers
- Advanced Applications of Euler’s Theorem
- Chinese Remainder Theorem for Large Systems of Equations
- Advanced GCD Algorithms for Multiple Numbers
- Efficient Algorithms for Finding All Divisors of a Number
- Understanding the Relationship Between GCD, LCM, and Divisors
- Generating and Using Primes Efficiently in Algorithms
- Pell’s Equation and Its Applications
- Diophantine Equations: Solving Integer Solutions
- Modular Multiplicative Inverses in Cryptography
- Fast Factorization Algorithms: Pollard’s Rho Method
- Factorization of Large Numbers Using Pollard's Rho and Elliptic Curves
- RSA Cryptography: Using Modular Arithmetic for Encryption
- Primality Testing with AKS Primality Test
- Discrete Logarithms and Their Computational Difficulty
- Shanks’ Baby-step Giant-step Algorithm for Discrete Logarithms
- RSA Key Generation and Security: Number Theory in Cryptography
- Efficient Use of Modular Inverses in Cryptographic Protocols
- Finding Modular Roots: Applications in Cryptography
- Fast Computation of Totients for Large Numbers
- Chinese Remainder Theorem and Polynomial Systems
- Generating Large Primes for Cryptographic Systems
- Understanding Fermat’s Little Theorem and Its Generalization
- Optimized Modular Exponentiation for Large Numbers
- Applications of Number Theory in Hash Functions
- Carmichael Numbers and Strong Pseudoprimes
- Factorization of Large Numbers Using Quadratic Sieve
- Understanding Fermat Pseudoprimes and Their Detection
- Number Theoretic Algorithms in Big Integer Arithmetic
- Sieve Algorithms for Counting Primes in Large Intervals
- Primes and Factorization in Polynomial Time
- Solving Modular Equations with Large Moduli
- Computing Modular Square Roots Using Tonelli-Shanks
- Efficient Algorithms for Finding Prime Factors of Large Numbers
- Mathematical Induction in Number Theory Proofs
- Primality Proving and the Role of Number Theory in Cryptography
- Advanced Techniques for Generating Prime Numbers
- Applications of Modular Arithmetic in Fast Fourier Transform
- Number Theoretic Optimizations for String Matching Algorithms
- Efficient Algorithms for Calculating Modular Inverses in Cryptography
- Advanced Number Theory in Computational Geometry
- Number Theoretic Algorithms in Graph Theory
- Finding Prime Divisors Using Pollard’s Rho Method
- Computing Large Integer Powers Using Modular Arithmetic
- Optimization Techniques for Modular Exponentiation in Competitive Programming
- Final Thoughts on Number Theory for Competitive Programming
This list starts with the basics of number theory, such as prime numbers, GCD, and LCM, and gradually progresses to advanced topics like modular exponentiation, cryptographic applications, primality testing, and solving Diophantine equations. Whether you're a beginner or an advanced practitioner, these chapters will cover everything you need to solve number-theory-based problems efficiently in competitive programming.