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.

2022-06-02math.GM

有无数对孪生素数

There are Infinitely Many Pairs of Twin Prime

我们证明了有无穷多对孪生素数。

2005-10-09math.GM

Infinitely many twin primes

In the proposed matrix primes, through which one can readily generate a sequence of primes. The paper also proposes a number of theorems proved by which an infinite number of prime numbers twins

2016-09-14math.GM

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.

2019-11-25math.GM

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…

2017-10-30math.GM

孪生素数问题的研究

Researches on the Twin Prime Problem

孪生素数问题主要是孪生素数的结构以及孪生孪生素数群是否存在无穷多个。本文通过构造一个特殊的簇数集（见文中公式（2.3）），证明了q的最小数的交集的前n列集合的集合数是孪生素数（q，q+2）的决定，且级数的最小数是发散的（见定理2），主要结果是定理2、定理3和定理4.孪生素数得到如此彻底的证明，必定有无穷多组孪生素数猜想。

2014-05-13math.GM

更多相关论文收起更多论文

关于某些特殊类型的素数的无限性

On the Infinitude of Some Special Kinds of Primes

本文的目的是试图建立一个通用模型的问题，多个多元数论函数同时表示素数的无穷多个整点。更具体地说，我们简要介绍了从欧几里德第二定理到格林-陶定理的研究背景--历史和现状。本文分析了整系数不可约一元多项式表示无穷多个素数的等价必要条件，找到了可能暗示费马素数只存在无穷多个的新的必要条件，得到了孙子剩余定理、广义欧拉函数、素数计数函数和Schinzel-Sierpinski猜想等的类比。这只是一个开始，还有很长的路要走。我们希望数论学家进一步考虑它。

2009-11-23math.GM

关于Polignac猜想的一种弱化形式

On a Weakened Form of Polignac Conjecture

Polignac [1]证明了对每个偶数自然数$2k (k\geq1)$，存在无穷多个连续素数$p_n$和$p_{n+1}$使得$p_{n+1}-p_n=2k$ .该猜想的弱化形式指出，对于每个$k\geq1$，存在无限多个素数$p$和$q$，使得$p-q=2k$。显然，波利尼亚克猜想的弱化形式意味着存在一个正整数$x_1,x_2,...,x_m,...$的无限序列，使得$x_1 (2k+x_1),x_2 (2k+x_2),...,x_m (2k+x_m)...$是两两互质的。在本文中，我们得到了一个比这个必要条件稍强的结果。这使得我们能够通过丰富的文献和大量的研究报告找到一些特殊的数论函数（如$2^x -1 $）的一个共同性质，这些函数可能表示无穷多个素数。但是，函数$2^{2^x}+1$没有此属性。这是否意味着费马素数的个数是有限的？Hardy和Wright [7]证明了费马素数的个数是有限的。但他们没有给出任何理由和解释。通过对费马数的因式分解，许多人认为[7]中的猜想成立。我们的工作能解释这种现象吗？我们将在另一篇论文中进一步讨论这个问题。基于我们的工作，可以给出一个新的充分条件，即存在无穷多个孪生素数（Sophie-Germain素数或Mersenne素数）。

2009-09-14math.GM

Simultaneous Elements Of Prescribed Multiplicative Orders

Let $u\ne \pm 1$, and $v\ne \pm 1$ be a pair of fixed relatively prime squarefree integers, and let $d\geq 1$, and $e \geq1$ be a pair of fixed integers. It is shown that there are infinitely many primes $p\geq 2$ such that $u$ and $v$ have simultaneous prescribed multiplicative orders $\text{ord}_pu=(p-1)/d$ and $\text{ord}_pv=(p-1)/e$ respectively, unconditionally. In particular, a squarefree odd integer $u>2$ and $v=2$ are simultaneous primitive roots and quadratic residues (or quadratic nonresidues) modulo $p$ for infinitely many primes $p$, unconditionally.

2021-03-05math.GM

Note On The Catalan Constant And Prime Triples

The existence of infinitely many consecutive prime triples $p_n$, $ p_{n+1}$, and $p_{n+2}$ as $n \to \infty$, is sufficient to prove that the Catalan constant $β(2)=0.9159655941\ldots $ is an irrational number. This note provides the detailed analysis. Moreover, the numerical data suggests that the irrationality measure is $μ(β(2))=2$, the same as almost every irrational real numbers.

2022-07-28math.GM