An Algebraic Approach to the Goldbach and Polignac Conjectures Using Mihailescu's Theorem and $p$-adic Analysis

We prove the Goldbach Conjecture using p-adic analysis and algebraic methods, requiring no knowledge of prime gaps or distribution by showing counterexamples exist if and only if certain polynomials have integer solutions. Assuming, for the sake of contradiction, a counter-example $2a$ exists, and labeling the set of primes up to $a$ as $\mathcal{P}$, we construct the Goldbach Polynomial $$\mathcal{G}_-(z) := \prod_{p_k \in \mathcal{P}} (z - p_k) - \prod_{p_k \in \mathcal{P}}p_k^{\alpha_k}$$ with conditions $\mathcal{G}_-(2a) = 0$ and all $\alpha_k$ are unique natural numbers. Using Hensel's Lemma, we prove each $2a - p_k$ must be a perfect prime power of only a prime in $\mathcal{P}$, giving solutions of the form $2a = p_j^{\alpha_j} + p_k$. Applying Mih\u{a}ilescu's Theorem (Catalan's Conjecture) shows the largest such polynomial is $$\mathcal{G}_-(z) = (z - 2)(z - 3) - 2^2 \times 3 : \mathcal{G}_-(6) = 0$$ proving no counterexamples exist for $a > 3$. We then prove the Goldbach Difference Conjecture similarly, from which the Polignac Conjecture follows.