有无数对孪生素数

On the Distribution of Twin Primes

We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter $λ_x$ and the estimation of the number of twin primes obtained as a result, is called a fundamental structure of the distribution of twin primes. Combining the latter with an asymptotic bound of $λ_x$, establishes venues, conducive to a discovery of a partial result that can be considered as a suitable variant of the prime number theorem. Furthermore, we obtain an asymptotic bound of the number of twin primes less than $x$.

A Possible Solution for Hilbert's Unsolved 8th Problem: Twin Prime Conjecture

We measure whether there are numerous pairs of twin primes (hereafter referred to as twin prime pairs) according to the prime number inferred by sieve of Eratosthenes. In this study, we reveal at least three additional twin prime pairs increased as n is increased by 1, while setting (6n+5)**2 as the range for estimating twin prime pairs. As a result, we prove the Twin Prime Conjecture proposed by de Polignac in 1849. That is, there are numerous twin prime pairs, indicating that there are numerous prime number p for each natural number k by making p + 2k as prime number for the case of k = 1.

On existence of infinite primes and infinite twin primes

The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The main tool of our investigation is a numeral system proposed recently that allows one to express various infinities and infinitesimals easily and by a finite number of symbols. The problems under consideration are the following and for all of them we give affermative answers: (i) do infinite primes exist? (ii) do infinite twin primes exist? (ii) is the set of i…

哥德巴赫猜想、孪生素数猜想及其它两个命题的证明

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$n^2+1$。

An equivalent problem to the Twin Prime Conjecture

In this short paper we will show, via elementary arguments, the equivalence of the Twin Prime Conjecture to a problem which might be simpler to prove. Some conclusions are drawn, and it is shown that proving the Twin Prime Conjecture is equivalent to proving that there cannot be an infinite string of consecutive natural numbers satisfying some specified equations.

经典Smarandache函数与孪生素数公式

The Classical Smarandache Function and a Formula for Twin Primes

本文利用经典的Smarandache函数给出了一个计算小于或等于x的孪生素数对的精确公式。本文还将该公式推广到素数对（p，p+2n）（n > 1）的计数.

Twin Primes In Quadratic Arithmetic Progressions

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the existence of infinitely many quadratic twin primes $n^{2}+1$ and $n^{2}+3$, $n \geq 1$, are proposed in this note.

Twin Prime Sieve

A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set of positive integers. Regularities of non-ranks allow gathering information on them to obtain a Legendre-type formula for the number of twin-ranks near primorial arguments.

Development of the matrix of primes and proof of an infinite number of primes-twins

This paper is devoted to the theory of prime numbers. In this paper we first introduce the notion of a matrix of prime numbers. Then, in order to investigate the density of prime numbers in separate rows of the matrix under consideration, we propose a number of lemmas and theorems that, together with the Dirichlet and Euler theorems, make it possible to prove the infinity of prime twins.

A Proof There Exists Infinitely Many Primes with a Gap of Exactly 2

This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a known twin prime pair, which can be mapped to a twin prime pair greater than the known one by multiplication. This function is shown to be unbounded and less than the true count of integers it seeks to approximate for the majority of twin prime pairs. Additionally, it is shown there must be infinitely many integers that map a twin prime pair to one larger tha…

Cycles and Patterns in the Sieve of Eratosthenes, Potential Twin Primes

This paper analyzes the emergence and distribution of potential twin primes, pairs of integers that are both relatively prime to the first n primes or to a given set M of primes, and which are the breeding grounds of true twin primes. It describes cyclical patterns of their location across the number line and provides a formula for counting them.

On the distribution of consecutive composite odd numbers and twin primes

We study pairs of consecutive odd numbers through a straightforward indexing. We focus in particular on twin primes and their distribution. With a counting argument, we calculate the limit of an alternating sum that is equal to 1 which means there are few twin primes. Finally, we show how to find the possible congruences for these prime numbers.

Infinitely many pairs of primes $p$ and $p+2$

We take the pre-sieved set to be all natural numbers $N=\{1,2,3,\dots\}$ with a sieve system:single sieve,double sieve,.... With single sieve, i.e. , remove out the multiple of a prime, we derive all the primes. With double sieve, i.e. , remove out the multiple and the multiple of a prime and $-2$ simultaneously, we get all the prime twins and prove that infinitely many prime twins exist under suitable conditions. Finally, with special 4 sieve, we prove that infinitely many prime twins exist without any restriction.

Properties and applications of the prime detecting function: infinitude of twin primes, asymptotic law of distribution of prime pairs differing by an even number

The prime detecting function (PDF) approach can be effective instrument in the investigation of numbers. The PDF is constructed by recurrence sequence - each successive prime adds a sieving factor in the form of PDF. With built-in prime sieving features and properties such as simplicity, integro-differentiability and configurable capability for a wide variety of problems, the application of PDF leads to new interesting results. As an example, in this exposition we present proofs of the infinitude of twin primes and the first Hardy-Littlewood conjecture for prime pairs (the twin prime number th…

孪生素数分布的集合论形式及其奇偶不平衡性

The Set-Theoretic Form of Twin Prime Distribution and Its Odd-Even Imbalance

本文用初等数论方法建立无遗漏包含所有孪生素数对的集合，通过其精确通解公式与奇偶不平衡性判断孪生素数对是否有限，并推广至波利尼亚克猜想。

有无穷多个表兄弟素数

There are infinitely many cousin primes

我们证明了有无穷多个表兄弟素数。

孪生素数问题例外集的一些结果

Some results in the exceptional set of Twin Prime Problem

本文给出了确定例外集中每个例外元的范围的新方法，基本上解决了孪生素数问题和哥德巴赫猜想问题。

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。

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。

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.

Theorems on twin primes-dual case

We prove dual theorems to theorems proved by author in \cite {5}. Beginning with Section 10, we introduce and study so-called "twin numbers of the second kind" and a postulate for them. We give two proofs of the infinity of these numbers and a sufficient condition for truth of the postulate; also we pose several other conjectures. Finally, we consider a conception of axiom of type "AiB".

筛法与孪生素数猜想

Sieve methods and the twin prime conjecture

本文提出一个关于孪生素数计数函数的新猜想，并证明了其在 $a=1$ 时的特例，从而确立了孪生素数猜想。

素数分布的线性性、合数的周期性、素数的混沌性、孪生素数的出现

Primes Distribution Linearity. Composite Numbers Cyclicity. Chaoticity of Primes. Twin Primes Occurrence

该研究提出素数分布的线性规律，证明所有素数位于96条半直线且满足对应线性关系，仅需1个线性方程和8个数即可判断合数，同时揭示自然数中合数出现的惊人周期性。

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…

论孪生素数的无穷性

On The Infinitude of the Twin Primes

本文基于 6x±1 素数表示定义孪生素数生成元，利用容许剩余类和改造的 Selberg 筛法划分自然数区间并估计候选生成元数量，指出奇偶性问题限制 Selberg 筛法，需寻求其他方法。

On Problems Related to Primes: Some Ideas

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their infinitude; establishment of a kind of similarity between natural numbers and numbers that appear in an arithmetic progression, similar formulae for primes and the so called generalized twin primes in an arithmetic progression and their infinitude; generalization of Bertrand postulate and a Bertrand like postulate for twin primes; some elementary implications of a si…

孪生素数问题的一种构造性启发式筛法

A Constructive Heuristic Sieve for the Twin Prime Problem

本文提出一种基于筛法理论的构造性启发式模型，通过$f(t;z)$函数分析将筛密度乘积表示为无穷级数之比，为近似Hardy-Littlewood孪生素数常数提供新路径。

A proof of the strong twin prime conjecture

For integers x and k, let T(x;2k) denote the number of twin prime pairs (p,p+2k) with a distance 2k<=2x**0.5 and p<=x (not p+2k<=x). Let Tg(x;2x**0.5) denote the average of T(x;2k) for all 2k<=2x**0.5. Logically, T(x;2k) should be a function of Tg(x;2x**0.5). We first, propose a sliding model to estimate Tg(x;2x**0.5). Second, derive the relations between T(x;2k) and Tg(x;2x**0.5) from the sieve structure. Third, settle the errors caused by the dependence of primes.

On a Stricter Twin Primes Conjecture, and on the Polignac's Conjecture in general

The Polignac's Conjecture, first formulated by Alphonse de Polignac in 1849, asserts that, for any even number M, there exist infinitely many couples of prime numbers P, P+M. When M = 2, this reduces to the Twin Primes Conjecture. Despite numerical evidence, and many theoretical progresses, the conjecture has resisted a formal proof since. In the first part of this paper, we investigate a stricter version of the conjecture, expressed as follows: ''Let $p_{n}$ be the n-th prime. Then, there always exist twin primes between $(p_{n}-2)^{2}$ and $p_{n}^{2}$ ''. To justify this conjecture, we formu…

Goldbach and Twin Prime Pairs: A Sieve Method to Connect the Two

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 …

On the existence of twin prime in an interval

Let $S_{(x,y]} = \left\{\frac{p_n}{p_{n+1}-2} :~ n\in I \right\}$, where $I = \left\{n :~ x<p_n \le y \right\}$, $p_n$ is the $n$-th prime and $x, y \in \mathbb{R}_{>0}$. If $M_α(x,y)$ denotes the $α$-power mean of the elements of $S_{(x,y]}$, it is shown that the existence of a twin prime pair in $(x,y]$ is implied if $\displaystyle \lim_{α\rightarrow \infty}M_α(x,y) > 1 - 2/y + O(y^{-2})$ for a sufficiently large $y$. For a special choice of $y$, we also find a lower bound for the mean: $\displaystyle \lim_{α\rightarrow \infty}M_α(x,x^β)>1-c/x^β+O(x^{-β}\log^{-1} x)$, where the constant $c>0…

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。

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 …

Structures in P based on Properties of Semigroup and Arithmetical Sequence H = (+-3*2; 1)

This paper presents results on structures in P based on tools developed from subjects of elementary number theory. Key findings are: The arithmetical sequence H = (+-3*2; 1) is in Z the smallest superset of P \ {3, 2}. H is a semigroup. A revised definition of P. Unique Gestalt of p in Z. The prime number lattice packing H exp n. The geometrical locus in HxH of the family of solutions of: - the set of prime twins, - the set of PRACHAR prime twins, - in H exp 2, H exp 3 family of solutions of the GOLDBACH conjunction. Partition of H in p exp 2-intervals. Prime numbers in p exp 2-intervals. Infi…

twin prime conjecture 的证明

A proof of the twin prime conjecture

本文通过开发通用方法证明了孪生素数猜想，展示了素数对的密度下界，并表明素数对的数目趋于无穷。

Prime-Generating Polynomial

We present a prime-generating polynomial $(1+2n)(p -2n) + 2$ where $p>2$ is a lower member of a pair of twin primes less than $41$ and the integer $n$ is such that $\: \frac {1-p}{2} < n < p-1$.