Goldbach Conjecture

In this paper we are going to give the proof of Goldbach conjecture by introducing a new lemma which implies Goldbach conjecture. By using Chebotarev-Artin theorem, Mertens formula and Poincare sieve we establish the lemma. The Goldbach conjecture was introduced in 1742 and has never been proven though it has been verified by computers for all numbers up to 19 digits. It states that all, even numbers above two are the sum of two prime numbers. All studies on Goldbach conjecture have failed. So we are going to give a complete proof of Goldbach conjecture. Let $ n $ an even integer such as above 20 and denote by $\mathbb{C}_{n}$ the set of the composite integers of $[1,n-1]$ to what we add 1 and let $f_{n}$ be the bijective mapping such that :$\begin{array}{ccc} f_{n}: \mathbb{C}_{n}& \rightarrow & n- \mathbb{C}_{n}\\ \text{ \ \ \ \ \ \ \ \ }m & \mapsto & n-m% \end{array}% $ \newline Denote by $G_{n}$ the subsect of $ n- \mathbb{C}_{n}$ consisting of prime numbers and $ G'_{n}$ that of composite numbers we have $n- \mathbb{C}_{n}=G_{n}\cup G'_{n}$ .Let $\mathcal{P}_{n}$ the set of prime numbers less than or equal to n . Let $$\delta(n)=card(G_{n}),\alpha(n)=card(\mathcal{P}_{n}\backslash G_{n}),\Pi(n)=card(\mathcal{P}_{n})$$ then $\Pi(n)=\delta(n) \alpha(n)$ ,obviously $\alpha(n)$ represents the number of ways to write n as the sum of two primes \\ \subsection{Lemma 1} $ \forall n\in \mathbb{N}$ , we have $\mathcal{P}_{n}\backslash G_{n}\neq \emptyset$