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

TL;DR

The paper proves the Goldbach and Polignac conjectures by employing p-adic analysis, algebraic methods, and Mihăilescu's theorem, establishing that counterexamples cannot exist for sufficiently large numbers.

Contribution

It introduces an algebraic framework combined with Mihăilescu's theorem to prove longstanding conjectures in number theory without relying on prime distribution.

Findings

Goldbach Conjecture is proven for all even integers greater than 3.

Polignac Conjecture is derived from the Goldbach Difference Conjecture.

Counterexamples to the conjectures do not exist beyond small cases.

Abstract

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.