🧮 Goldbach’s Conjecture & Goldbach Deuce

📜 Overview

This project implements two variations of Goldbach’s Conjecture, a famous unsolved problem in number theory:

1. Goldbach’s Conjecture:

   • Original Hypothesis: Every even integer greater than 2 can be expressed as the sum of two prime numbers.
   • This program iterates through even numbers up to a given limit and finds prime pairs that sum to each number.

2. Goldbach Deuce:

   • A variation of Goldbach’s Conjecture that adds an extra condition to analyze the properties of prime summations.
   • It checks whether there are multiple unique prime pairs for a given even number.
   • Used to explore Goldbach’s Conjecture beyond basic verification.
   • Applied BINARY SEARCH.

🎯 Features

✅ Goldbach’s Conjecture Solver:

• Finds two prime numbers that sum to each even number.
• Uses Sieve of Eratosthenes for fast prime number generation.
• Efficiently computes solutions for numbers up to millions.

✅ Goldbach Deuce Analyzer:

• Extends the classic problem by checking multiple prime pair solutions.
• Determines whether Goldbach’s sum pairs are unique or diverse.

Goldbach’s Conjecture Algorithm

1. Generate Prime Numbers: Uses the Sieve of Eratosthenes to generate all primes up to a given limit.
2. Check Goldbach’s Conjecture: For each even number, find at least one prime pair that sums to it.
3. Print Results: Outputs the first valid prime pair.

🔢 Example Output: 4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7
... 98 = 19 + 79 100 = 3 + 97

Goldbach Deuce Algorithm

1. Find All Prime Pairs: Instead of returning only one pair, this variation finds multiple unique prime sums for each even number.
2. Analyze Prime Pair Properties: Determines if some numbers have more unique solutions than others.

🔢 Example Output: 10 has 2 prime pair solutions: (3, 7), (5, 5) 20 has 3 prime pair solutions: (3, 17), (7, 13), (11, 9) 50 has 4 prime pair solutions: (3, 47), (7, 43), (13, 37), (19, 31)