First Prime in Euler's Number 🧮

The Billboard Puzzle 🤔

The mysterious mathematical billboard that appeared in Silicon Valley

This project solves old mathematical puzzle that appeared on a billboard in Silicon Valley. The billboard showed a URL in the format:

{first 10-digit prime found in consecutive digits of e}.com

The Solution 💡

The answer is 7427466391.com. Our program finds this number in approximately 0.002 seconds!

How it Works 🔧

The solution combines several mathematical and computational techniques:

1. Generating e's Digits

   • Uses scaled integer arithmetic for perfect precision
   • Implements the series: e = 2 + 1/1! + 1/2! + ...
   • Adds extra precision digits for accuracy

2. Prime Testing

   def is_prime(number):
       if number < 2: return False
       if number == 2: return True
       if number % 2 == 0: return False
       
       for i in range(3, int(number ** 0.5) + 1, 2):
           if number % i == 0: return False
       return True

3. Sliding Window Search

   • Efficiently searches through consecutive digits
   • Converts digit sequences to numbers
   • Tests each number for primality

Results 📊

First 10-digit prime in e: 7427466391
Time taken: 0.0023488998413085938 seconds

Features ✨

• Pure Python implementation
• No external dependencies
• Fairly fast execution
• Well-covered test suite
• Well-documented code
• Mathematical elegance

Usage Example 📝

from euler_prime import find_first_prime_in_e

# Find first 10-digit prime in e
result = find_first_prime_in_e(10)
print(f"First 10-digit prime in e: {result}")

Testing 🧪

Run the tests using pytest:

python -m pytest tests/

Project Structure 📁

.
├── README.md
├── euler_prime.py     # Main implementation
├── tests/
│   └── test_euler.py # Test cases
├── requirements.txt   # Project dependencies
└── LICENSE           # MIT License

The Math Behind It 📚

Euler's number (e) is a mathematical constant approximately equal to 2.71828... It can be represented as an infinite series:

e = 2 + 1/1! + 1/2! + 1/3! + 1/4! + ...

Our implementation uses this series along with scaled integer arithmetic to generate precise digits of e.

Performance Optimization 🏃‍♂️

The solution employs several optimizations:

• Early termination in primality testing
• Efficient digit extraction
• Smart precision handling
• Sliding window optimization

License 📄

This project is licensed under the MIT License - see the LICENSE file for details.

Author ✍️

[Danil Zanozin][https://github.com/danilmezor]

Acknowledgments 🙏

• Google for creating this puzzle
• The mathematical beauty of Euler's number
• The open-source community

---

Made with ❤️ and Python