Exploring the Goldbach Conjecture Through Number Theory and the Pigeonhole Principle

The Goldbach Conjecture, one of the oldest unsolved problems in number theory, states that every even integer greater than two can be expressed as the sum of two prime numbers. In this paper, the problem is solved through a combination of number-theory methods and the pigeonhole principle. By fragmenting the conjecture into smaller, provable components, a logical framework is built up that connects the circulation of odd primes to the structure of even numbers. This method accentuates how elementary number theory, when paired with combinatorial judgment, can provide a clear pathway toward understanding why such pairs of primes must exist.