Proof of the Goldbach Conjecture by extension to the negative number line and using a probabilistic approach

ABSTRACT. The Goldbach conjecture is one of the oldest unsolved problems in number theory and in the whole mathematical field. It is firstly mentioned by Prussian mathematician Christian Goldbach in his correspondence to Swiss mathematician Leonard Euler almost 300 years ago in 1742. The conjecture states that any even number greater than 2 could be build as sum of 2 prime numbers. Although there exist some proofs for heavily modified versions of the (odd) Goldbach conjecture, until this day, no proof exist for the original, 2-prime even Goldbach conjecture. In this paper after we have established some crucial statements around the conjecture and created a search algorithm to find the pairs of primes for any even numbers, we need to extend the Goldbach conjecture to the negative integer domain to be more able to use an important attribute of primes that they do exist in infinite numbers in both the positive and negative integer domain before we could proof the conjecture in its more generic and extended form using a probabilistic method.