A Proof of "Goldbach's Conjecture"
Roger Ellman
“哥德巴赫猜想”通过分析奇素数的所有组合如何成对求和，产生所有的偶数来证明。
"Goldbach's Conjecture" proven by analysis of how all combinations of the odd primes, summed in pairs, generates all of the even numbers.
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Effective prime factors and Goldbach's Theorem/Conjecture
根据有效素因子分布的相似性定理，用两部分法证明了哥德巴赫定理，从而证明了哥德巴赫猜想。
The Proof of Goldbach Conjecture, Twin Primes Conjecture and Other Two Propositions
通过创造一种新的方法，证明了世界著名的难题哥德巴赫猜想、孪生素数猜想、命题（C）和命题$n^2+1$。
Geometrical Equivalents of Goldbach Conjecture and Fermat Like Theorem
给出了哥德巴赫猜想的五个几何等价式，其中一个称为类费马定理。
In this paper we propose an alternative formulation of the binary and ternary Goldbach conjectures as the systems of equations involving the Euler $ϕ$-function.
Can We Prove Goldbach's Conjecture?
本文提出了一种证明哥德巴赫猜想的策略：每个大于2的偶数都可以写成两个素数之和。
On Goldbach's Conjecture
证明了如果每个奇数$n > 5$是三个素数之和，则每个偶数$n > 2$是两个素数之和.在Cramér猜想的基础上，给出了Goldbach猜想的一个条件证明.关于哥德巴赫猜想的理论和实验结果允许在条件证明中使用比克莱默猜想限制性更小的猜想。一个基本的结果迈尔的文件Cramér的模型是批评。
Goldbach's conjecture
管理员说明：由于使用假名，违反了arXiv政策，被arXiv管理员撤回。
Let $p_{1}$, ..., $p_{k}$ be the first $k$ odd primes in succession. Let $n$ be an even integer such that $n > p_{k}$. We conjecture that if none of $n - p_{1}$, ..., $n - p_{k}$ are prime, then at least one of them has a prime factor which is greater than or equal to $p_{k}$. In this brief note, we observe that Goldbach's conjecture follows from this conjecture.
In this paper we introduce a simple method of searching for the prime pairs in the famous Goldbach Conjecture. The method, which is based on certain integer identities as well as an observation related to the remainder property, enables us to cut the search length by half and therefore can be considered as a short cut in searching for the prime pairs in the Goldbach Conjecture.
The problem of the least prime number in an arithmetic progression and its applications to Goldbach's conjecture
等差数列的最小素数问题是数论中的重要课题之一。在[11]中，我们首次研究了这个问题与哥德巴赫猜想之间的关系。本文进一步考虑了它在哥德巴赫猜想中的应用，改进了文[11]中的结果。此外，我们还试图推广等差数列中的最小素数问题，并给出哥德巴赫猜想的一个类比。
Goldbach Conjecture and First-Order Arithmetic
利用自然数的双曲分类、本质区域和哥德巴赫猜想函数的概念，证明了一阶算术中哥德巴赫猜想的一个证明的存在性意味着另一个证明的存在性，这个证明在一定的扩展中并不适用于通过适当的动态过程产生的与自然数相关的所有时间状态.
The Goldbach conjecture that, every even integer is the sum of two primes, has been open since 1742. This paper details a road map to a proof of Goldbachs conjecture based on a function that estimates the number of Goldbach pairs. It is shown that the estimate of the number of Goldbach pairs increases as the even integer increases and that the error term is small. Further detailing the functions, assumptions and error term may finally lead to a proof of Goldbachs conjecture.
Hyperbolic classification of natural numbers and Goldbach Conjecture
通过一个动力学过程，我们给出了依赖于时间的偶数的无限集中的哥德巴赫猜想的一个特征。
Goldbach Conjecture and the least prime number in an arithmetic progression
本文研究了哥德巴赫猜想与等差数列中最小素数的关系。给出了哥德巴赫猜想的一个新的弱化形式。我们证明了这个弱化形式和Chowla假设的一个弱化形式意味着每一个足够大的偶数可以写成两个不同素数之和。哥德巴赫猜想与算术级数中的最小值关系的研究综述在本文中，我们对哥德巴赫猜想与算术级数中的最小值关系进行了研究。我们给了一个新的形式，可能来自哥德巴赫猜想。我们认为，这一事件和乔拉假设的一个事件意味着，整个两个国家都可以像两个不同的总理一样，充分利用。
We answer the question positively. In fact, we believe to have proved that every even integer $2N\geq3\times10^{6}$ is the sum of two odd distinct primes. Numerical calculations extend this result for $2N$ in the range $8-3\times10^{6}$. So, a fortiori, it is shown that every even integer $2N>2$ is the sum of two primes (Goldbach conjecture). Of course, we would be grateful for comments and objections.
Proof of the Goldbach conjecture in a more stringent form
利用人工（p '，n'）系统证明了偶数>（p（x））^2是两个p > p（x）的和。
From the Goldbach Conjecture to the Theorem
本文证明了欧拉于1742年提出的哥德巴赫猜想“所有大于大于2的正偶数都可以表示为两个素数之和”是正确的。这个结果部分地基于Wilson定理，并且补充了我们将问题转换为丢番图方程的推理。后者是几何证明的主方程。
A proof of Goldbach's conjecture that all even numbers greater than four are the sum of two primes
在本文中，我介绍了一个模型，它允许一个证明哥德巴赫假说。该模型是通过研究哥德巴赫划分所显示的一个倒置的镜像的所有素数，直到一些偶数等于最后一个素数加三。然后，模型的下半部分以2为一步向右移动，这展示了下一个偶数的哥德巴赫分割。只要模型包含所有的素数，直到得到的偶数减3，然后哥德巴赫假设可以证明，如果它可以证明，每一步必须产生一个哥德巴赫划分，直到一个到达下一个素数加1。我证明了情况一定是这样的。
About Goldbach strong conjecture
本文利用数族的分类方法，对6n+1和6n+5（其中n为整数）进行分类（这类族包含所有大于3的奇素数和其它与素数有关的复合数）。我们将使用这种分类，以尝试一种方法来解决哥德巴赫强猜想。通过二进制数带的几何方法，我们设想了一种研究所述问题的新形式。
I define Goldbach counting function with N > 0 and square-free P > 0. Decomposition of this function is discovered and deduction formula is found. I propose a hypothesis on upper bound of Goldbach counting function and prove that Goldbach conjecture is true under this hypothesis. Also, I propose a second hypothesis on upper bound of Goldbach counting function and prove that the twin prime conjecture is true under this hypothesis
The binary Goldbach conjecture is also true
本文在前期证明的基础上估计正整数 $K_α$，从而推出每个大于 $4$ 的偶数均可表示为两个素数之和，证明二元哥德巴赫猜想成立。
Some considerations in favor of the truth of Goldbach's Conjecture
本文给出了关于哥德巴赫猜想的一些思考。这项工作是基于基本结果的数论，它提供了一个建设性的方法，允许，给定一个偶数，找到至少一对素数根据GC。最后给出了这种构造方法与一类运筹学问题之间的联系。
On a Proof of the Goldbach Conjecture and the Twin Prime Conjecture
本文利用纽结上的连通和运算证明了纽结与数之间存在一一对应的关系。在这个关系中，素数结被双射地分配有素数，使得素数2对应于三叶结。从这个关系，我们有一个分类表的结，其中结是一对一的编号分配。此外，用于编号$2^n$的第$n$诱导步骤的该分配由用于先前$n-1$步骤的该分配确定。从这个归纳法中，我们可以解决一些数论中的问题，如哥德巴赫猜想和孪生素数猜想。
The Strong Goldbach Conjecture: Proof for All Even Integers Greater than 362
强哥德巴赫猜想可以追溯到1742年。它指出，任何大于大于4的偶数都可以写成两个素数的和。从那时起，没有人能够证明这个猜想。到目前为止，唯一已知的结果是，每个足够大的偶数N都可以写成一个素数与至多两个素数的乘积之和。此外，该猜想已被证明对所有直到4.10 ^18的偶数都成立。本文证明了该猜想对所有大于大于362的偶数都成立。
The Goldbach conjecture resulting from global-local cuspidal representations and deformations of Galois representations
在Langlands整体程序的基本一般框架下，由p次根覆盖的局部p-adic椭圆半模构造了对应于局部（左）尖点形式的局部p-adic椭圆半模.在相同的背景下，全球和本地的双线性变形的伽罗瓦表示诱导其各自的剩余域的不变性，以及全球和本地的双线性量子变形留下不变的顺序的惯性子群。更具体地说，导致其分裂的闭曲线的逆量子形变直接导致哥德巴赫猜想。
This article is a survey based on our earlier paper ("The 'Vertical' Generalization of the Binary Goldbach's Conjecture as Applied on 'Iterative' Primes with (Recursive) Prime Indexes (i-primeths)" [11]), a paper in which we have proposed a new generalization of the binary/"strong" Goldbach's Conjecture (GC) briefly called "the Vertical Goldbach's Conjecture" (VGC), which is essentially a meta-conjecture, as VGC states an in finite number of Goldbach-like conjectures stronger than GC, which all apply on "iterative" primes with recursive prime indexes (named "i-primes"). VGC was discovered by t…
We approach a new proof of the strong Goldbach's conjecture for sufficiently large even integers by applying the Dirichlet's series. Using the Perron formula and the Residue Theorem in complex variable integration, one could show that any large even integer is demonstrated as a sum of two primes. In this paper,the Riemann Hypothesis is assumed to be true in throughout the paper. A novel function is defined on the natural numbers set.This function is a typical sieve function.Then based on this function,several new functions are represented and using the Prime Number Theorem,Sabihi's theorem, an…
The Hardy-Littlewood conjectures on the twin primes and the binary Goldbach problem are true
该研究通过有限Ramanujan展开，无条件证明了孪生素数计数函数的Hardy-Littlewood渐近公式，并以类似方法解决了任意偶数可表示为两素数之和的共轭猜想。
Proof of Goldbach's Conjecture
在定义了某些称为Range和Row的自然数子集之后，我们假设（1）存在一个可以产生素数的函数，以及（2）每个大于2的偶数，如A，可以表示为n个素数的和。我们通过DC（A）小于或等于n来证明这一点。每个人都有一个相似的地方，每个人都有一个相似的地方，每个人都有一个相似的地方。证明了在任意一行中，对于任意大于2的偶数，DC（A）=2，即每个大于2的素数都是两个素数之和.从而证明了哥德巴赫猜想。
In this article we study in depth the Dirichlet theorem, which states that if a, b are relative prime integers, the sequence p = an + b contains infinite prime numbers, we simplify and generalize this theorem, we enunciate some special theorems (from Elias), we see equivalence with Bertrand's postulate and we deduce the validity of the Goldbach conjecture. Finally we enunciate some additional results that are deduced from the conjecture-theorem of Goldbach-
Associate a unique numerical sequence called the modular signature with each positive integer, using modular residues of each integer under the prime numbers, and distinguishing between the core seed primes and non-core seed primes used to create the modular signatures. Group the modular signatures within primorials. Use elementary sieve properties and combinatorial principles to prove the twin primes and Goldbach conjectures.
Definitive Proof of Goldbach's conjecture
本文通过推导近似素数对数量的方程并进行误差分析，证明对于所有n > 622的偶数素数对数量大于1，从而确定性证明了哥德巴赫猜想。
On Partitions of Goldbach's Conjecture
利用素数定理和启发式概率方法，导出了哥德巴赫猜想的一个近似剖分公式。哥德巴赫猜想的一个强形式以哥德巴赫猜想的划分的下界函数的形式出现。数值计算表明，分区的上下边界函数满足一个简单的函数方程。假设该不变缩放性质对于所有偶数整数$n$成立，则下限和上界可以表示为简单的指数。
ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a procedure for the determination of the upper bounds for prime gaps different from the most famous and known approaches. The proposed method analyzes the distribution of prime numbers using the set of relative numbers. Using negative numbers too, it becomes intuitive to understand that that the arrangement of 2P+1 consecutive numbers that goes -P to P, is the only a…
Is it possible that the Goldbach's and Twins primes conjectures are true with an algebraic approach?
本文通过为每个大于2的m构造偏序集同构并赋予阿贝尔群运算，用代数方法刻画偶数并定位满足猜想的数对，从而论证哥德巴赫猜想与孪生素数猜想成立。
La cometa di Goldbach e ... le altre
哥德巴赫彗星是哥德巴赫函数g（n）在区间[3，N]中的曲线，其中N是一个大的正整数。函数g（n）计算2n可以表示为两个奇素数之和的不同方式的数量。一个帐户，希望令人满意的和可访问的，给出了层，使彗星。通过数论的几个（有时是历史性的）结果，出现了类似于哥德巴赫的其他理论。这些与奇正整数序列有关，类似但不完全是素数序列。
Goldbach's Rule
哥德巴赫猜想，“任何大于大于2的偶数都可以表示为两个素数之和”，因为它不可能是其他的，所以被重新命名为哥德巴赫法则。证明了这个猜想，证明了素数对加到任何大于大于2的偶数上的存在性是小于那个偶数的素数序列存在性的自然副产品。首先，它表明，其余的取消过程中，确定素数小于一个偶数也calluders素数对添加到该偶数作为一个自然的一部分的过程。然后，一个最小限度的素数对添加到一个偶数表示的偶数和存在的每一个偶数大于2。此外，证明中使用的推理和公式被证明对观察持有。
An Algebraic Approach to the Goldbach Conjecture
本文利用p进分析和Mihailescu定理，通过构造多项式和代数方法证明哥德巴赫猜想，并推导出波林亚克猜想。
Matrix Online Characteristic Number and Its applications in Goldbach Conjecture,Polignac Conjecture,the Twin Prime Conjecture
本文共分三章，第一章由连续奇数决定，研究了一类矩阵序列的内在性质。通过建立矩阵在线数的概念、特征和列在线数的运用数学归纳法证明了这类矩阵在线数的一些性质特征（定理1）。最后，给出了哥德巴赫猜想的一个尝试证明（定理6）。这是作者在多年探索素数性质的过程中的研究和发现，并认为这一发现具有重要意义。在第二章中，定义了矩阵主示性数和矩阵主特征序列的概念（定义1）。首先证明了任意偶数至少在两个群上可表示为两个奇素数之差（定理4）。其次，我们证明了存在无穷多个被4隔开的奇素数（定理9）。最后，证明了如果对任意自然数m，S（3）与s（2 m +3）的交大于1，则存在无穷多个以2 m为间隔的奇素数（定理11）。在第三章中，主要是由于任意奇数自然数a，S（a）与s（a+2）的交集不是空集，且集合中的数远大于1，其中S（a）={k，若2k+a是素数，k是自然数}，P是素数集合，N是自然数集合。证明了孪生素数存在无穷多个，从而解决了数论中的孪生素数问题。
Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture
前k个素数的乘积m_k（2.. p_k）有邻居m_k +/- 1，且所有素除子都在p_k之外，这意味着有无穷多个素数[Euclid]。p_k和m_k之间的素数都在半群Z_{m_k}（.）模M_k乘法的一个例子。由于无平方模，Z_{m_k}是2^k个群的不交并，具有尽可能多的幂等元-每个m_k的除数一个幂等元，它们形成布尔格BL。研究了Z_{m_k}的生成元及其格的可加性.证明了BL中的每一个互补对都加1 mod m_k，BL中的每一个偶幂等元e在G_1中都有后继e+1.因此，G_1+G_1 \equiv E是Z_{m_k}中偶数剩余的集合，所以每个偶数剩余是两个单位根之和，证明了“剩余的哥德巴赫”mod m_k（“GR”）。. . .通过扩展mod m_k的剩余量并“携带”权重m_k的a < p_{k+1}来对k进行归纳，得到整数的素筛。哥德巴赫猜想（“GC”）对某个2n的失败与GR（k）对某个k的矛盾。根据伯特兰公设（在素数i<p 1时<2i for each i>），连续的2n在重叠的区间内，而G_1 mod m_k中的最小合成单位是p_{k+1} ^2，得到“GC”：每个2n > 4是两个奇素数的和。
Sieve Method and Landau Problem
利用筛法解决了哥德巴赫猜想和孪生素数猜想等Landau的四个不可攻问题。
n 1937 Ivan Vinogradov proved the three prime sum version of the Goldbach Conjecture, often called the weak form of Goldbach Conjecture. And that it holds for "sufficiently large" odd natural numbers. In this work we use Dirichlet Theorem, Modulo Arithmetics, etc. to extended Vinogradov's Theorem such that every sufficiently large natural number (both even and odd) can be expressed as a sum of three primes. We highlight the configuration of primes for any special case of the three prime sum. Hence we obtain Vinogradov's Theorem as a special case of this extended version. We show how Vinogradov…
This paper proposes, and demonstrates the efficacy of, an elementary method for establishing a lower bound for cardinalities of selected sets of twin primes, and shows that the proof employed may be modified for selected sets of Goldbach pairs. Our sieve method is centred on the restrictive properties of intervals, specifically regarding divisibility distributions. We implicitly use the Chinese Remainder Theorem by way of the use of the midpoint in our intervals, and consider the sieve of Eratosthenes in such a way as to find a set of primes whose distribution is mirror-symmetrical about that …
Conjectures on Partitions of Integers as Summations of Primes
本文给出了一类关于整数分拆为素数和的猜想，它们是哥德巴赫猜想的推广。
Fillings method in number theory
用一种新方法发展了数论中的若干结果。加强形式下的哥德巴赫猜想就是其中之一。
Empirical verification of a new generalisation of Goldbach's conjecture up to $10^{12}$ (or $10^{13}$) for all coefficients $\leq 40$
本文测试了第一作者提出的 Goldbach 猜想新推广（GGC），将其验证至 $10^{12}d$ 或部分至 $10^{13}d$（系数 $\leq 40$），比较了四种验证算法的运行时间，并提出了 GGC 的扩展以推广孪生素数猜想。
Weak Golbach's Conjecture from Isomorphic and Equivalent Odd Prime Number Functions
数学家们一直试图通过添加素数来证明弱哥德巴赫猜想，就像猜想中所说的那样。然而，我们认为，解决方案不需要通过分析来解决。我们没有尝试添加素数来证明猜想，而是开发了一个素数函数P_{odd}（x）p>2，包括奇数素数p > 2，同构和等价于一个函数N_{odd}（x）n>1，包括大于1的奇数自然数，n_{odd} > 1，其中，它的三个元素的和导致奇数大于7，证明了猜想。
In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some interesting results related to the densities of sequences. The method is based on the direct construction of the Eratosthenes-type double sieve and does not use empirical and heuristic reasoning.
In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some interesting results related to the densities of sequences. The method is based on the direct construction of the Eratosthenes-type double sieve and does not use empirical and heuristic reasoning.
Study on Hilbert's Eighth Problems
本文通过筛函数变换改进筛法与算术数列素数定理，宣称证明广义黎曼猜想和黎曼猜想，并改进塞尔伯格线性筛法以证明哥德巴赫猜想与孪生素数猜想，从而基本解决希尔伯特第八问题。
In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. For any prime pairs $p>5$ and $ q>5 $, construct a disjoint infinite set sequence $A_1, A_2, \ldots, A_i. \ldots $, such that the number of prime pairs ($p_i$ and $q_i $, $p_i-q_i = p-q $) in $A_i $ increases gradually, where $i>0$. So twin prime conjecture is true. We also prove that for any even integer $m>2700$, there exist more than 10 prime pairs $(p,q)$, such that $p+q=m$. Thus Goldbach conjecture is true.
The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea of our approach is summarized on the creation and on the analyzing sequence of sets of distinct co-primes with the first $n$ primes, $\left\{ p_i :\, i\leq n \right\}$, and the important properties of the modulus linear combination of the co-prime sets, $H=\left(1,p_{n+1},..., Π_{i=1}^n p_i-1\right) $, that gives sets of even numbers $\{0,2,4,..., Π_{i=1}^n…
Conjectures in number theory
本文在探讨素数重要性的基础上，提出了两个关于特定形式整数（$1+3.2^{20n}$ 和 $1+3.2^{4+20n}$）在特定模条件下非素数的数论猜想。
In this paper proof of the twin prime conjecture is going to be presented. Originally very difficult problem (in observational space) has been transformed into a simpler one (in generative space) that can be solved. It will be shown that twin primes could be obtained through two stage sieve process, and that will be used to show that exist infinite number of twin primes. The same approach is used to prove the Polignac's conjecture for cousin primes.
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