A combinatorial proof framework for Goldbach's conjecture
哥德巴赫猜想的一个组合反证框架

摘要

本文提出了一种基于组合数学和解析数论相结合的哥德巴赫猜想证明方法。通过构建奇数序列的严格配对系统，并引入改进的素数分布估计，我们证明了每个不小于六的偶数都可以表示为两个奇素数之和。主要创新点包括：首先建立三色配对模型，将配对严格分类为CC型、PC型和PP型（C表示合数，P表示素数），其次提出配对密度函数φ(m)来量化分析素数分布，最后应用改进的Rosser-Schönfeld不等式导出猜想不成立的精确矛盾。理论分析和数值验证表明，该方法不仅提供了图解的数学证明，为探索素数分布的深层规律提供了新的方法论支撑，还为相关数论问题的运用研究提供了新的思路，包含猜想在强化大数据加密强度的运用，以及DNA/RNA大数据测序工程中运算加速或精度提高等运用。

Abstract

This paper proposes a method to prove the Goldbach conjecture based on the combination of combinatorial mathematics and analytic number theory. By constructing a rigorous pairing system for odd sequences and introducing an improved estimation of prime distribution, we prove that every even number not less than 6 can be expressed as the sum of two odd primes. The main innovations are as follows: first, establishing a three-color pairing model, which rigorously classifies pairings into CC type, PC type, and PP type (where C denotes a composite number and P denotes a prime number); second, proposing a pairing density function φ(m) to quantitatively analyze prime distribution; finally, applying the improved Rosser-Schönfeld inequality to derive the precise contradiction regarding the invalidity of the conjecture.Theoretical analysis and numerical verification show that this method not only provides a graphical mathematical proof, offering new methodological support for exploring the underlying laws of prime distribution, but also presents new ideas for research on the applications of related number theory problems. These applications include the use of the conjecture in enhancing the encryption strength of big data, as well as in accelerating computations or improving accuracy in big data sequencing projects for DNA/RNA, among others.