Added Pell's Equation solver in C++

[shepherdking67]

📌 Pell’s Equation

This program solves the Diophantine equation:
x^2 - D*y^2 = 1
where D is a non-square positive integer.

Example

Input: D = 61
Output: x = 1766319049, y = 226153980

✅ Contribution

• Added pells_equation.cpp under mathematical/.
• Finds the fundamental solution (x, y) using continued fraction expansion of √D.

[github-actions]

💎 Code Quality Check Results

❌ Missing Complexity Analysis

These files don't include time/space complexity:

• CPP/algorithms/mathematical/pells_equation.cpp

Required: Add comments explaining time and space complexity (e.g., Time: O(n log n), Space: O(n))

❌ Missing Algorithm Description

These files don't explain what the algorithm does:

• CPP/algorithms/mathematical/pells_equation.cpp

Required: Add a description explaining the algorithm, its purpose, and how it works

📚 Quality Standards

To maintain high quality, every contribution should include:

1. ✍️ Algorithm Description

   • Explain what the algorithm does
   • Describe the approach and methodology
   • Include use cases or applications

2. 📊 Complexity Analysis

   • Time complexity (e.g., O(n log n))
   • Space complexity (e.g., O(n))
   • Brief explanation of why

3. 💬 Meaningful Comments

   • Explain complex logic
   • Document function parameters
   • Add inline comments for clarity

4. ✅ Test Cases/Examples

   • Demonstrate the code works
   • Show different input scenarios
   • Include edge cases

💡 Example Template

"""
Binary Search Algorithm

Description: Searches for a target value in a sorted array using divide-and-conquer

Time Complexity: O(log n) - halves search space each iteration
Space Complexity: O(1) - only uses constant extra space
"""

def binary_search(arr, target):
    # Initialize pointers
    left, right = 0, len(arr) - 1

    while left <= right:
        mid = (left + right) // 2

        # Check if target found
        if arr[mid] == target:
            return mid
        # Search right half
        elif arr[mid] < target:
            left = mid + 1
        # Search left half
        else:
            right = mid - 1

    return -1  # Not found

# Test cases
if __name__ == "__main__":
    test_arr = [1, 3, 5, 7, 9]
    print(binary_search(test_arr, 5))  # Output: 2
    print(binary_search(test_arr, 6))  # Output: -1

🔧 How to Fix

1. Review each file mentioned above
2. Add the missing documentation
3. Push your changes
4. The workflow will re-run automatically

💪 You've Got This!

These checks help maintain quality and make your contribution more valuable to learners. Thank you for taking the time to improve! 🙏

---

Quality over quantity - let's build something amazing together! 🌟

[github-actions]

🎉 Welcome to Hacktoberfest 2025, @shepherdking67! 🎃

Thank you for your first contribution to our DSA repository! Here's what happens next:

🔍 Automatic Checks

• ✅ Code Validation: Passed
• 🧪 Compilation Tests: Passed

📋 Next Steps

🎯 Great job! Your code compiled successfully. Maintainers @Karanjot786 and @Pradeepsingh61 will review your PR soon.

🎁 What You Get

• 🏆 Hacktoberfest Credit: This PR counts toward your 6 PR goal for exclusive T-shirt + Tree!
• 🌟 Hall of Fame: You'll be featured in our contributors list
• 📚 Learning: Code review feedback from experienced developers

💡 Tips for Success

• Follow our Contributing Guidelines
• Add comments explaining your algorithm
• Include time/space complexity analysis
• Test your code before submitting

Welcome to the community! 🚀

[github-actions]

🤖 Automated PR Status

🔍 Code Validation

✅ Passed - File naming and structure look good!

🧪 Compilation Tests

✅ Passed - All code compiles successfully!

📋 Overall Status

🎉 Ready for Review - Your PR has passed all automated checks!
👥 Maintainers have been notified for review.

---

This comment was generated automatically. Checks will re-run when you push new commits.

[Karanjot786]

@Pradeepsingh61 merge it