Kaida Shi
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 $n^2+1$.
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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.
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 …
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。
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
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 …
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.
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…
We solve Landau's four unattackable problems, including Goldbach Conjecture and Twin Prime Conjecture through sieve method.
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…
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.
Study on Hilbert's Eighth Problems
本文通过筛函数变换改进筛法与算术数列素数定理，宣称证明广义黎曼猜想和黎曼猜想，并改进塞尔伯格线性筛法以证明哥德巴赫猜想与孪生素数猜想，从而基本解决希尔伯特第八问题。
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.
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。
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…
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