A Proof of the Goldbach and Polignac Conjectures

Abstract: This paper will give both the necessary and sufficient conditions required to find a counter-example to the Goldbach Conjecture by using an algebraic approach where no knowledge of the gaps between prime numbers is needed. To eliminate ambiguity the set of natural numbers, $\mathbb{N}$, will include zero throughout this paper. Also, for any sufficiently large $a \in \mathbb{N}$ the set $\mathcal{P}$ is the set of all primes $p_i \leq a$. It will be shown there exists a counter-example to the Goldbach Conjecture, given by $2a$ where $a \in \mathbb{N}{> 3}$, if and only if for each prime $p_i \in \mathcal{P}$ there exists some unique $\alpha_i \in \mathbb{N}$ and a mapping $\mathcal{G}-:\mathbb{C} \to \mathbb{C}$ where \begin{equation*} \mathcal{G}-(z) = \prod{p_i \in \mathcal{P}}(z - p_i) - \prod_{p_i \in \mathcal{P}}p_i^{\alpha_i} : \mathcal{G}_-(2a) = 0. \end{equation*} A proof of the Goldbach Conjecture will be given utilizing Hensel's Lemma and Catalan's Conjecture showing that $a = 3$ is the largest solution and no counter-examples exist. A similar method will be employed to give the necessary and sufficient conditions when an even number is not the difference of two primes with one prime being less than that even number. A proof will then be given that every even number is the difference of two primes by utilizing Hensel's Lemma and Catalan's Conjecture showing that $a = 3$ is the largest solution and no counter-examples exist. A proof of the Polignac Conjecture will then follow.