Goldbach and Twin Prime Pairs: A Sieve Method to Connect the Two
Tom Milner-Gulland
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20 Pages. This paper is complemented by a proof of the Riemann hypothesis by the same author, found at arXiv:1805.06746 . The key steps and intuitions behind the work are explained in the introductory sections immediately following the abstract, together with the section Introduction to Theorems 2 and 3
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 midpoint. Bounds are established through the use of the formulae closely associated with the Prime Number theorem and the Mertens theorem. We show that the Goldbach conjecture is true if the Riemann hypothesis is true.
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.
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…
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
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…
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 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.
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。
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…
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.
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 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.
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 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…
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…
