An Approach Using Combinatorics for the Decomposition of Odd Pairs in Relation to Goldbach's Conjecture

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. Although our results do not set up the Goldbach conjecture, they give our perspective into the combinatorial mechanisms underlying Goldbach-type decompositions and clarify limitations that any probable disproof must fulfill.