Can We Prove Goldbach's Conjecture?

Goldbach's conjecture

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An elementary conjecture which implies the Goldbach conjecture

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.

A Simple Method for searching for Prime Pairs in the Goldbach 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

The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we further consider its applications to Goldbach's conjecture and refine the result in [11]. Moreover, we also try to generalize the problem of the least prime number in an arithmetic progression and give an analogy of Goldbach's conjecture.

哥德巴赫猜想与一阶算术

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.

Goldbach's Racetrack

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.

自然数的双曲分类与哥德巴赫猜想

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

In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened form of the Chowla Hypothesis imply that every sufficiently large even integer may be written as the sum of two distinct primes. Résumé La conjecture de Goldbach et le plus petit nombre premier dans une progression arithmétique Dans ce document, nous essayons d'étudier les relations entre la conjecture de Goldbach et le plus petit nombre premier dans une progre…

Is Goldbach Conjecture true?

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.

哥德巴赫猜想的更严格证明

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。

从哥德巴赫猜想到定理

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。

哥德巴赫猜想所有大于4的偶数都是两个素数之和的证明

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。

关于哥德巴赫强猜想

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。

A Hypothesis on Upper Bound of Goldbach Counting Function

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

二元哥德巴赫猜想同样成立

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

This paper presents some considerations about the Goldbach's conjecture (GC). The work is based on elementary results of the number theory and it provides a constructive method that permits, given an even integer, to find at least a pair of prime numbers according to the GC. Eventually a correlation will be provided between this constructive method and a class of problems of operations research.

关于哥德巴赫猜想和孪生素数猜想的一个证明

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

The Strong Goldbach conjecture dates back to 1742. It states that every even integer greater than four can be written as the sum of two prime numbers. Since then, no one has been able to prove the conjecture. The only best known result so far is that every sufficiently large even integer N can be written as the sum of a prime number and the product of at most two prime numbers. Additionally, the conjecture has been verified to be true for all even integers up to 4.10^18. In this paper, we prove that the conjecture is true for all even integers greater than 362.

由整体-局部尖点表示和伽罗瓦表示变形引起的哥德巴赫猜想

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。

A "Vertical" Generalization of the binary Goldbach's Conjecture as applied on primes with prime indexes of any order i (i-primes)

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…

An approach towards the proof of the strong Goldbach's conjecture for sufficiently large even integers

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…

关于孪生素数与二元哥德巴赫问题的Hardy-Littlewood猜想成立

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。

哥德巴赫猜想的证明

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。

El teorema de Dirichlet, la conjetura de Goldbach y algunas consecuencias

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-

Proofs of the Twin Primes and Goldbach Conjectures

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.

哥德巴赫猜想的确定性证明

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。

关于哥德巴赫猜想的分拆

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。

The importance of finding the upper bounds for prime gaps in order to solve the twin primes conjecture and the Goldbach conjecture

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…

是否可能用代数方法证明哥德巴赫猜想与孪生素数猜想为真？

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

Goldbach's comet is the plot of the Goldbach function g(n), in the interval [3, N], with a large positive integer N. The function g(n) counts the number of different ways in which 2n can be expressed as the sum of two odd primes. An account, hopefully satisfying and accessible, is given for the layers that make up the comet. By means of several (sometimes historical) results of Theory of Number, other conjectures, similar to the Goldbach's one, emerge. These are related with sequences of odd positive integers, like but not quite to the prime sequence.

哥德巴赫法则

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。

利用Mihailescu定理和p进分析的代数方法研究哥德巴赫与波林亚克猜想

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

This article consists of three chapters.In Chapter 1, it is determined by the consecutive odd numbers, and study to the intrinsic properties of a class of matrix sequence. Through the establishment of matrix online number concept, characteristics and the online number column use mathematical induction to prove the some properties of this kind of matrix on the number of online features (Theorem 1). Finally, it is given a trial to prove the Goldbach conjecture (Theorem 6). This is the author in the years to explore prime properties in the process of research and discovery, and believe that this …

Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture

The product m_k of the first k primes (2..p_k) has neighbours m_k +/- 1 with all prime divisors beyond p_k, implying there are infinitely many primes [Euclid]. All primes between p_k and m_k are in the group G_1 of units in semigroup Z_{m_k}(.) of mutiplication mod m_k. Due to the squarefree modulus Z_{m_k} is a disjoint union of 2^k groups, with as many idempotents - one per divisor of m_k, which form a Boolean lattice BL. The generators of Z_{m_k} and the additive properties of its lattice are studied. It is shown that each complementary pair in BL adds to 1 mod m_k and each even idempotent …

筛法与Landau问题

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。

An Extension Of Vinogradov's Theorem

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…

关于整数分拆为素数和的猜想

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。

数论中的填充方法

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。

对所有系数 $\leq 40$ 的 Goldbach 猜想新推广至 $10^{12}$（或 $10^{13}$）的经验验证

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

Mathematicians has been trying to prove the weak Goldbach's conjecture by adding prime numbers, as stated in the conjecture. However, we believe that the solution does not need to be analytically solved. Instead of trying to add prime numbers to prove the conjecture, we developed a prime number function P_{odd}(x)p>2, including odd primes p > 2, isomorphic and equivalent to a function N_{odd}(x)n>1, including odd natural numbers greater than one, n_{odd} > 1, in which, the sum of three of its elements result in odd numbers greater than 7, proving the conjecture.