Add Sieve of Eratosthenes algorithm implementation #[HACKTOBERFEST 2025]

[piyushkumar0707]

🚀 Overview

This PR implements the Sieve of Eratosthenes - an ancient, highly efficient algorithm for finding all prime numbers up to a given limit, with multiple optimizations and variants.

✨ Features

• ✅ Classic sieve implementation - The traditional O(n log log n) algorithm
• ✅ Optimized version - Only processes odd numbers after 2
• ✅ Segmented sieve - Find primes in specific ranges efficiently
• ✅ Prime counting - Count primes without storing them
• ✅ Performance comparison - Benchmarks against trial division
• ✅ Multiple variants - Educational and production-optimized versions

🎯 Why This Matters

The Sieve of Eratosthenes is fundamental for:

• Cryptography - RSA key generation requires large primes
• Number theory research - Prime distribution studies
• Algorithm education - Classic example of optimization techniques
• Mathematical computing - Efficient prime generation for calculations
• Competitive programming - Fast prime generation in contests

📚 Implementation Details

• Time Complexity: O(n log log n) - significantly faster than trial division
• Space Complexity: O(n) for the boolean sieve array
• Optimizations:
  • Skip even numbers (except 2)
  • Start marking from p²
  • Segmented approach for large ranges

🔧 Advanced Features

• sieve_of_eratosthenes() - Basic implementation
• sieve_of_eratosthenes_optimized() - Odd-numbers-only optimization
• segmented_sieve() - Memory-efficient for large ranges
• count_primes_sieve() - Count without storage
• Performance benchmarking - Speed comparison demonstrations

🧪 Testing & Examples

Comprehensive testing including:

• Small prime sets with manual verification
• Large scale performance tests (up to 10,000+)
• Edge cases (n < 2, n = 2, n = 3)
• Segmented sieve range testing
• Speed comparisons with trial division

📊 Performance Benefits

Example: Finding primes up to 1,000

• Sieve method: ~0.001 seconds
• Trial division: ~0.05+ seconds
• Speedup: 50x+ faster

📁 Files Added

• mathematics/sieve_of_eratosthenes.r - Complete implementation with variants
• Updated DIRECTORY.md with new entry

🎓 Educational Value

Perfect for learning:

• Algorithm optimization techniques
• Trade-offs between time and space complexity
• Historical algorithms still used today
• Performance analysis methods

🎃 Hacktoberfest 2025

Adds a classical, highly-optimized algorithm with significant practical applications in cryptography and computational mathematics.

[Copilot]

Pull Request Overview

This PR implements the Sieve of Eratosthenes algorithm with multiple variants and optimizations. The implementation provides a classical prime number generation algorithm that's significantly more efficient than trial division methods.

Key changes:

• Complete Sieve of Eratosthenes implementation with basic and optimized variants
• Segmented sieve for memory-efficient range-based prime finding
• Performance comparison utilities and comprehensive testing examples

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 2 comments.

File	Description
mathematics/sieve_of_eratosthenes.r	Main implementation with multiple sieve variants, performance benchmarks, and comprehensive examples
DIRECTORY.md	Added entry for the new sieve algorithm file

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[siriak]

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