A Proof of the Goldbach and Polignac Conjectures

For any prime $p_i \leq a$, where $a \in \mathbb{N}$ and $a > 3$, it is possible to define some $b_i \in \mathbb{N}$ where $a - b_i = p_i$. Using this fact, it will be shown that there exists some $q_i \in \mathbb{N}$ where $a + b_i = q_i$, and $q_i$ is a prime number, by proving solutions cannot exist when $2a = q_i +p_i$ for all prime $p_i \leq a$, and $\prod_{i = 1}^{\pi(a)}q_i = \prod_{i = 1}^{\pi(a)}p_i^{\alpha_i}$ for some $\alpha_i \in \mathbb{N}$ when $a > 3$. A similar method will be employed to prove every even number is the difference of two primes by assigning to any odd, prime $p_i \leq a$ some $b_i' \in \mathbb{N}$ where $a - b_i' = -p_i$, along with the existence of $q_i' \in \mathbb{N}$ where $a +b_i' = q_i'$ and showing solutions cannot exist for $a > 3$ when $2a = q_i' - p_i$ for all odd $p_i \leq a$, and $\prod_{i = 2}^{\pi(a)}q_i' = \prod_{i = 2}^{\pi(a)}p_i^{\alpha_i'}$ for some $\alpha_i' \in \mathbb{N}$. The Polignac Conjecture will then follow.