¶ Prime Numbers and Sieve Algorithms
Here are 100 chapter titles on Prime Numbers and Sieve Algorithms, progressing from beginner to advanced, tailored for competitive programming:
I. Foundational Concepts (20 Chapters)
- What are Prime Numbers?
- Divisibility Rules and Prime Factorization
- Checking Primality: Basic Approach (Trial Division)
- Limitations of Trial Division
- Optimizations for Trial Division (Checking up to sqrt(n))
- Prime Factorization using Optimized Trial Division
- Introduction to Sieve Algorithms
- The Sieve of Eratosthenes: Basic Implementation
- Understanding the Sieve of Eratosthenes
- Optimizations for the Sieve of Eratosthenes
- Time and Space Complexity Analysis of Sieve of Eratosthenes
- Applications of Sieve of Eratosthenes
- Generating Primes within a Range
- Counting Primes within a Range
- Precomputing Primes using Sieve
- Introduction to Prime Number Theorems (Brief Overview)
- Distribution of Prime Numbers
- Practice Problems: Basic Primality Testing
- Practice Problems: Prime Factorization
- Practice Problems: Sieve of Eratosthenes (Basic)
II. Intermediate Techniques (30 Chapters)
- Segmented Sieve
- Understanding Segmented Sieve
- Implementing Segmented Sieve
- Space Optimization in Segmented Sieve
- Sieve of Atkin
- Understanding Sieve of Atkin
- Implementing Sieve of Atkin
- Comparison of Sieve Algorithms (Eratosthenes, Atkin, Segmented)
- Applications of Segmented Sieve
- Finding Prime Factors using Sieve
- Euler's Totient Function (Phi Function)
- Calculating Phi Function using Sieve
- Mobius Function
- Calculating Mobius Function using Sieve
- Prime Counting Function (Pi(n)) - Approximation
- Legendre's Formula for Prime Counting (Brief Overview)
- Practice Problems: Segmented Sieve
- Practice Problems: Sieve of Atkin
- Practice Problems: Euler's Totient Function
- Practice Problems: Mobius Function
III. Advanced Concepts and Applications (30 Chapters)
- Advanced Sieve Techniques
- Wheel Factorization
- Optimizations with Wheel Factorization
- Sieve of Sundaram
- Sieve of Brun
- Applications in Cryptography (RSA, etc. - Brief Overview)
- Primality Testing: Miller-Rabin Test (Probabilistic)
- Primality Testing: AKS Primality Test (Deterministic - Brief Overview)
- Finding Large Prime Numbers
- Generating Random Prime Numbers
- Prime Gaps
- Twin Primes
- Prime Triplets
- Arithmetic Progressions of Primes
- Case Study: Solving Real-World Problems with Prime Numbers and Sieves
- Competitive Programming Strategies for Prime Number Problems
- Optimizing Prime Number Code for Speed and Memory
- Testing and Debugging Strategies for Prime Number Implementations
- Prime Number Problem Solving Techniques: Pattern Recognition
- Prime Number Problem Solving Techniques: Problem Decomposition
IV. Expert Level and Competitive Programming Challenges (20 Chapters)
- Advanced Prime Number Problem Sets (Codeforces, LeetCode, etc.)
- Hard Level Prime Number Problems and Solutions
- Contests and Challenges: Prime Number Focus
- Analyzing Time and Space Complexity of Advanced Prime Number Algorithms
- Advanced Optimization Techniques for Prime Number Problems
- Parallel Processing with Prime Number Algorithms (if applicable)
- Distributed Prime Number Computation (if applicable)
- Implementing Prime Number Algorithms in Different Programming Paradigms
- Performance Tuning of Prime Number Implementations
- Advanced Debugging and Profiling of Prime Number Code
- Code Review and Best Practices for Prime Number Implementations
- Prime Numbers and System Design (Rarely Applicable Directly)
- Research Topics in Prime Numbers
- The Future of Prime Number Research
- Prime Numbers and Machine Learning (Indirectly Related)
- Prime Numbers and Artificial Intelligence (Indirectly Related)
- Mastering Prime Numbers for Competitive Programming Success
- Connecting Prime Numbers to Other Number Theory Problems
- Exploring Variations of Prime Number Problems with Different Constraints
- Applying Prime Numbers to Complex Real-World Scenarios
- Prime factorization algorithms (Pollard's Rho, etc.)
- Discrete logarithm problem and its relation to prime numbers
- Elliptic curve cryptography and its connection to prime numbers
- Primality proving algorithms
- Integer factorization algorithms
- Lattice-based cryptography and its relation to prime numbers
- Quantum computing and its impact on prime number related problems
- Open research problems in number theory related to prime numbers
- The Riemann Hypothesis and its connection to prime numbers
- Distribution of prime numbers in arithmetic progressions (Dirichlet's Theorem)
- Prime number sieves for special types of numbers (e.g., quadratic primes)
- Applications of prime numbers in cryptography (beyond RSA)
- Applications of prime numbers in coding theory
- Applications of prime numbers in hash table design
- Prime numbers and their role in random number generation
- Prime numbers and their connection to other mathematical areas
- History of prime number research
- Famous unsolved problems related to prime numbers
- The beauty and fascination of prime numbers
- The ongoing quest to understand prime numbers and their properties.