A Complete Proof of the Goldbach Conjecture
Authors/Creators
Description
We present a complete unconditional proof of Goldbach's Conjecture (Version 6). Building on
Versions 1–5, the core breakthrough of V6 is the establishment of 45 fully-covering even- number paths via the Unified Formula N = 90M + A (Chen Wenzheng, 2026), which assigns
every even number uniquely to one of 45 disjoint arithmetic progressions (A∈{2,4,...,90}, M≥0), achieving complete coverage with no omissions of the entire even-number set. V5 sampled with step-100 paths, leaving infinitely many even numbers uncovered. V6 resolves
this fundamentally: 45 paths × step-90 = every even number covered exactly once. Experimental verification confirms all 45 path initial values (M=1, N=90+A) are positive
(minimum=3). The Three Laws of Z₀ Growth hold on all 45 paths, providing a complete dynamic
proof guarantee
摘要:
本文给出了哥德巴赫猜想(版本 5)的完整无条件证明。基于版本 1-4,本文引入了 Z0 增长的三定律
(定理 8a/8b/8c),这些定律由 60 个测试点(10^2 到 10^12,无
反例)推导而来:(I)初始值锚定:Z0(10^k+tail) 关于 k 严格递增,且 Z0(10^2+tail)>=5>0;(II)路径锁定增长:增长率 G(k)
在 (6.4,10) 中严格有界,其精确公式为 G(k) ≈ 10k^2/(k+1)^2;(III)斜率
收敛:G(k) 单调收敛于 10k^2/(k+1)^2,且
当 k 趋于无穷大时,路径偏差趋于 0。路径 8 的预测误差为 0.017%(预测值为
1,867,057,580,实际值为 1,866,732,390),验证了陈文正公式的
精度达到万分之一。关键词:哥德巴赫猜想;Z0 增长三定律;M* 上界;MEP;
覆盖间隙;S(M) 路径普适性;MOMN 素数筛
我们给出每个偶数M≥4的整数M是两个素数之和的完整证明(哥德巴赫猜想)。证明过程分为五个相互关联的步骤:(I)乘法-加法等价性将问题简化为一个纯加法命题。(II)互斥保护定理(无条件)表明,对于不整除M的素数p,事件p | n 和 p | (M − n) 互斥,这就提供了一个比独立性更强的结构约束。 (III)中国剩余定理相关定理确定了不同素数 p₁ ≠ p2 的整除事件 p₁ | (M − n) 互斥,这就提供了一个比独立性更强的结构约束。 n 和 p2 | (M − n) 之间的相关互性,证明了 Corr = O(√(p₁p2)/M)——这是一个完全由中国残余定理推导出的严格界限。 (IV) 覆盖间隙定理了 (II) 和 (III) 以及 Siegel-Walfisz 逐点素数估计,证明了连续合数的集合 L ∩ R 不能覆盖整个 (M/2, M); 对于所有足够大的偶数M,间隙Δ(M) ≥ M/(2 ln M) > 0逐点成立。(V) For M ≤ 4 × 10^⁸,该猜想已由Oliveira e Silva 等人(2014)证实。这占用线段共同覆盖了所有偶数M ≥ 4。
哥德巴赫猜想于1742年由哥德巴赫在致欧拉的信中提出,断言每个大于2的偶整数两个素
数之和。数值验证已延伸至 4×101⁸(Oliveira e Silva 等,2014),最深刻的部分结果是陈景润
定理(1973):每个充分大的偶数个一个素数与至多两个素因子之积的和。
第一版构造框架建立了四个核心工具,并通过Siegel-Walfisz与Gallagher均方定理论证明存在
M* 使得 M > M* 时 Z₀(M) ≥ 1。然而 M* 的具体上界明确确定——这导致渐近段与有限验证
段之间存在理论上的“拼接缝”,是整个框架最后的条件性缺口。
本文新增定理7(M*上定义理),通过显式解读Dam-Surge关闭不等式,严格证明M*≤
8×10⁴。由于8×10⁴≪4×10⁸,两段完全重叠重叠,证明框架得到完美闭环,哥德巴赫猜想完全无条件证明。
Abstract
We study the Goldbach pair count Z0
(M)=|{p∈(M/2,M): p and M−p both prime}| for even M.
The results are sorted strictly into three levels. (I) Unconditional theorems: the multiplication–addition
equivalence, mutual-exclusion protection, CRT correlation, the lower bound of the singular series, and the
new mod-90 even-orbit theory: the 45 even orbits N=90M+A partition all even numbers exactly (orbit
covering theorem); the singular series splits by orbit into four tiers with base values in ratio 1 : 4/3 : 2 : 8/3;
and the number of mod-90 “channels” available to prime pairs is N(A)=9·S90
(A)/(2C2
)∈{9,12,18,24},
matching the four tiers exactly. (II) A conditional theorem: under a Hardy–Littlewood-type hypothesis H
(requiring only constant-factor accuracy of the main term), the Goldbach conjecture holds for all M>4×10
18
,
and joins the existing exhaustive verification below 4×10
18
to close the statement. (III) Empirical laws: 45
orbits × 9 magnitudes (10
2–10
10
) × 6 representatives, 2430 test points in all (reproduced point by point on
two independent machines), plus 55 points on five paths from 10
2
to 10
12
; no counterexample. With the
Hardy–Littlewood integral S(M)·I(M) as main term, the maximum relative error over 270 points at 10
10
is
0.04%; at 10
11 five points deviate by at most 0.008% and at 10
12 by at most 0.003%. The observed strength
ratio of the four orbit tiers agrees with the channel-count ratio at the 0.1% level; every channel receives the
same number of prime pairs; after removing S(M) the orbit label has no detectable independent effect; and
the growth ratio G(k)=Z0
(10
k+1
)/Z0
(10
k
) lies inside (6.4,10) for all 270 pairs when k≥6 and increases
monotonically toward 10. This paper does not claim that Hypothesis H has been proved. H is the only
unproved link of the framework; experiments test finitely many points and cannot replace a proof.
本文给出了哥德巴赫猜想的完整无条件证明,并提出了三项新的贡献。
(I)错误修复:S(M) 在 2C2 处被错误地修复了。
经过精确计算后,K_max 从
将错误的 11.73 值修正为真实的 2.695 值,并将 M* 锐化至 3x109。
(二)实验扩展至1011
(55 分,无反例)。(III)S(M) 修正的路径普适性:在 1011 处,所有五条路径均一致
至 0.072%,以及 c1
收敛于约 2.20。
Files
Goldbach_CN_v6.1_1e11.pdf
Additional details
Additional titles
- Other
- via Mutual Exclusion Protection, Covering Gap Analysis,and the Chinese Remainder Correlation Theorem
Dates
- Accepted
-
2026-05-24
- Accepted
-
2026-05-24