An Algebraic Approach to the Goldbach Conjecture

Jason R. South

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arXiv · 2206.01179

An Algebraic Approach to the Goldbach Conjecture

Jason R. South

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Abstract

We prove the Goldbach Conjecture using p-adic analysis and algebraic methods, requiring no knowledge of prime gaps or distribution. To begin, we define the set of primes up to $a \in \mathbb{N}$ as $\mathcal{P}_a$. It will be shown that if a counter-example $2a$ exists, there exists a polynomial \[ \mathcal{G}_-(z) = \prod_{p_k \in \mathcal{P}_a} (z - p_k) - \prod_{p_k \in \mathcal{P}_a}p_k^{\alpha_k} \] with conditions $\mathcal{G}_-(2a) = 0$ and all $\alpha_k \in \mathbb{N} \cup \{0\}$. We then use Hensel's Lemma to assist in proving this polynomial structure places the constraint that the solutions for $2a$ require \[ S_a = \{p_i \in \mathcal{P}_a : 2a = p_i^{\alpha_i} + p_i \} \] which is impossible for $a > 3$ as this requires $a\# \; |\; 2a$.

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Jason R. South. 2022-05-31. An Algebraic Approach to the Goldbach Conjecture. https://arxiv.org/abs/2206.01179

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