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 $x$.
We present a deterministic relationship between relative primes and twin primes in successively larger sequences of the natural numbers. This enables setting a finite lower limit on the occurrence of actual twin primes in an unbounded selection of domains.
Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and estimation of the number of primes on the twin primes rows are obtained. A new method of twin prime infinity is proposed.
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.
Twin prime number problem is mainly the structure of the twin prime numbers and whether there are infinitely many prime twins group. In this paper, by constructing a special cluster number set(see formula(2.3)in the paper), proves that the number of set number of the first n columns set the intersection of the minimum number of q is decision of the prime twins (q, q+2), and the minimum number of series is divergent(see Theorem 2).The main rezults are Theorem 2,Theorem 3 and Theorem 4.Prime twins so thoroughly proved there must be infinitely many groups of twin prime conjecture.
This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is a twin prime. Twin primes are primes having a prime gap of ${2}$. The pairs ${(5,7),(11,13),(41,43)}$, etcetera are all twin primes. This paper envisions that if the proposed system of inequalities can be proven to have infinite solutions, the Twin Prime Conjecture will evidently be proven true. The paper also derives a novel upper bound on the prime gap, $…
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
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.
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…
By creating a new method, the author proved the well-known world's baffling problems Goldbach conjecture, twin primes conjecture, the Proposition (C) and the Proposition $n^2+1$.
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.
This short paper presents an exact formula for counting twin prime pairs less than or equal to x in terms of the classical Smarandache Function. An extension of the formula to count prime pairs (p, p+2n), n > 1, is also given.
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.
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.
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.
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.
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.
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…
In the paper, there are new found methods to determine the range of every exceptional element in exceptional set, we can solve Twin primes problem and Goldbach Conjecture problem basically.
A celebrated conjecture of Hardy and Littlewood provides with an asymptotic formula for the counting function of the twin primes. We give an unconditional proof of such a formula by means of a finite Ramanujan expansion of the counting function expressed in terms of the von Mangoldt function and its incomplete form. In a completely analogous way, we solve the conjugate conjecture on the representations of any even integer as the sum of two prime numbers.
In this paper we use the connected sum operation on knots to show that there is a one-to-one relation between knots and numbers. In this relation prime knots are bijectively assigned with prime numbers such that the prime number 2 corresponds to the trefoil knot. From this relation we have a classification table of knots where knots are one-to-one assigned with numbers. Further this assignment for the $n$th induction step of the number $2^n$ is determined by this assignment for the previous $n-1$ steps. From this induction of assigning knots with numbers we can solve some problems in number th…
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.
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".
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…
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…
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…
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.
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…
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.
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 …
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…
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 …
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…
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the subset of Germain primes PG = {p and 2p + 1 are primes}, and the subset of quadratic primes Pf = {p = n^2 + 1 primes}. These subsets of primes are widely believed to be infinite subsets of prime numbers.
1. There is no existing any quadratic interval $η_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $η_{n}$ goes averagely linear with n to infinity. 2. The exact law of the number $π(n)$ of prime numbers smaller or equal to n is given. As an approximation of that we get the prime number theorem of Gauss for great values of n. 3. We derive partition laws for $π(η_{n})$, for the number of twin primes $π_{2}(η_{n})$ in quadratic intervals $η_{n}$ and for the multiplicity $π_{g}(2n)$ of representations of Goldbach-pairs for a given even number 2…
This article presents a necessary and sufficient theorem for N numbers, coprime two by two, to be prime simultaneously. It generalizes V. Popa's theorem, as well as I. Cucurezeanu's theorem, Clement's theorem, S. Patrizio's theorems, etc. Particularly, this General Theorem offers many characterizations for twin primes, for quadruple primes, and so on.
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 of the more general dePolignac conjecture on the existence of infinitely many primes pairs p and p + 2k, k => 1, is proposed in this note.
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.
All the known approximations of the number of primes pi(n) not exceeding any given integer n are derived from real-valued functions that are asymptotic to pi(x), such as x/log x, Li(x) and Riemann's function R(x). The degree of approximation for finite values of n is determined only heuristically, by conjecturing upon an error term in the asymptotic relation that can be seen to yield a closer approximation than others to the actual values of pi(n) within a finite range of values of n. By considering the density of each of the set of (i) all integers n, (ii) Dirichlect integers n = a+md, and (i…
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbe…
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…
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.
