An Algebraic Approach to the Goldbach and Polignac Conjectures

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 $q_i, \alpha_i \in \mathbb{N}$ where $a < q_i < 2a$ and $2a = q_i + p_i$ along with the condition that $\prod_{p_i \in \mathcal{P}}q_i = \prod_{p_i \in \mathcal{P}}p_i^{\alpha_i}$.A substitution of $q_i = 2a - p_i$ for each $q_i$ from each sum gives the product relationship $\prod_{p_i \in \mathcal{P}}(2a - p_i) = \prod_{p_i \in \mathcal{P}}p_i^{\alpha_i}$.Therefore, if a counter-example exists to the Goldbach Conjecture, then there exists a mapping $\mathcal{G}_-:\mathbb{C} \to \mathbb{C}$ where $$\mathcal{G}_-(z) = \prod_{p_i \in \mathcal{P}}(z - p_i) - \prod_{p_i \in \mathcal{P}}p_i^{\alpha_i}$$ and $\mathcal{G}_-(2a) = 0$. A proof of the Goldbach Conjecture will be given utilizing Hensel's Lemma to show $2a$ must be of the form $2a = p_i^{\alpha_i} + p_i$ for all primes up to $a$ when $a > 3$. However, this leads to contradiction since $2a < a\#$ for all $a > 4$.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. To begin, let $a \in \mathbb{N}_{> 3}$ with the condition that the function $\gamma(a + 1)$ is equal to one if $a + 1$ is prime and zero otherwise. $2a$ is a counter-example if and only if for each prime $p_i \in \mathcal{P}$ there exists some unique $u_i, \beta_i \in \mathbb{N}$ where $2a < u_i \leq 3a$ and $2a = u_i - p_i$ along with product relationship $\prod_{p_i \in \mathcal{P}}u_i = (a + 1)^{\gamma(a + 1)}\prod_{p_i \in \mathcal{P}}p_i^{\beta_i}$. A substitution of $u_i = 2a + p_i$ for each $u_i$ from each sum gives $\prod_{p_i \in \mathcal{P}}(2a + p_i) = (a + 1)^{\gamma(a + 1)}\prod_{p_i \in \mathcal{P}}p_i^{\beta_i}.$ Therefore, if a counter-example exists, it is possible to define the mapping $\mathcal{G}_+ : \mathbb{C} \to \mathbb{C}$ where $$\mathcal{G}_+(z) = \prod_{p_i \in \mathcal{P}}(z + p_i) - (a + 1)^{\gamma(a +1)} \prod_{p_i \in \mathcal{P}}p_i^{\beta_i}$$ and $\mathcal{G}_+(2a) = 0$. A proof will then be given that every even number is the difference of two primes by showing $2a$ must be of the form $2a = p_i^{\beta_i} - p_i$ for all odd primes up to $a$ when $a > 3$ to the equation above, leading to the same contradiction as the Goldbach Conjecture since $2a < a\#$ for $a > 4$. These proofs will have implications for proving the Polignac Conjecture.