In this paper, a rigorous proof of the strong Goldbach Conjecture is provided. This proof is mainly based on application of Chebotarev -Artin theorem ,Mertens formula ‘s . By using the principle of inclusion -exclusion of Moivre in the set which elements are in arithmetic progression We show that it is always possible to find at least one pair of prime numbers according to the validity of the condition [1]. Moreover, the prime numbers theorem is used to investigate the number of primes’ pairs corresponding to even numbers. The obtained results showed that all even numbers have at least one pair of primes verifying this conjecture. Thus, this result provides a rigorous proof for Goldbach conjecture which is still considered, to our best knowledge, among the open problems of mathematics.

Keywords: Goldbach conjecture, Chebotarev-Artin theorem, Mertens formula, Moivre