Goldbach Conjecture Solver

This Python script calculates and verifies Goldbach's Conjecture for all even numbers up to a user-defined target sum. The results are saved in a text file called Goldbach-conjecture.txt.

Overview

Goldbach's Conjecture is an unproven theory in mathematics which states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This script implements the Sieve of Eratosthenes algorithm to generate a list of prime numbers and then tests the conjecture for all even numbers up to a specified target sum.

Features

• Prime Number Generation: Uses the Sieve of Eratosthenes to efficiently generate all prime numbers up to the target sum.
• Goldbach Conjecture Verification: For every even number from 4 to the target sum, the script checks if it can be expressed as the sum of two primes.
• File Output: The results are written to a file called Goldbach-conjecture.txt, with each line showing an even number as the sum of two prime numbers.

How It Works

1. Sieve of Eratosthenes: The sieve_of_eratosthenes function generates all prime numbers up to the given target sum.
2. Goldbach Conjecture: The goldbach_conjecture function iterates through all even numbers from 4 to the target sum. For each even number, it checks if it can be written as the sum of two primes from the list generated by the sieve.
3. Output: The results are saved to Goldbach-conjecture.txt in the format even_number = prime1 + prime2.

Usage

1. Clone the repository or download the script.

2. Run the script using Python:

   python goldbach_conjecture.py

3. Enter the target sum when prompted:

   Total to be sought>>

4. The results will be written to Goldbach-conjecture.txt.

Example

If you input 20 as the target sum, the script will verify the conjecture for all even numbers from 4 to 20, producing an output similar to the following in Goldbach-conjecture.txt:

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7
12 = 5 + 7
14 = 3 + 11
16 = 3 + 13
18 = 5 + 13
20 = 3 + 17

Requirements

• Python 3.x

No additional libraries are required.

License

This project is licensed under the MIT License.

Acknowledgments

• The Sieve of Eratosthenes algorithm for generating prime numbers.
• Goldbach's Conjecture, an unsolved problem in number theory that continues to intrigue mathematicians.