A Strengthened Form of the Strong Goldbach Conjecture

Abstract

This paper proves a strengthened form of the strong Goldbach conjecture by showing that its negation is unprovable in ZFC, assuming this theory is sound. We reformulate the conjecture using an infinite set and show, based on properties of this set, that the assumption that there is a proof of its negation leads to a contradiction. Whereas the traditional approaches focus on the control over the distribution of the prime numbers by means of circle method and sieve theory, the proof is based on the constructive role of the primes within the natural numbers, reflecting their multiplicative character.