It is shown that if every odd integer $n > 5$ is the sum of three primes, then every even integer $n > 2$ is the sum of two primes. A conditional proof of Goldbach's conjecture, based on Cramér's conjecture, is presented. Theoretical and experimental results available on Goldbach's conjecture allow that a less restrictive conjecture than Cramér's conjecture be used in the conditional proof. A basic result of the Maier's paper on Cramér's model is criticized.
According to the similarity theorem on the distributions of the effective prime factors and by using two-part method, Goldbach theorem and, consequently, Goldbach conjecture was proved.
In this paper we propose an alternative formulation of the binary and ternary Goldbach conjectures as the systems of equations involving the Euler $ϕ$-function.
Let $p_{1}$, ..., $p_{k}$ be the first $k$ odd primes in succession. Let $n$ be an even integer such that $n > p_{k}$. We conjecture that if none of $n - p_{1}$, ..., $n - p_{k}$ are prime, then at least one of them has a prime factor which is greater than or equal to $p_{k}$. In this brief note, we observe that Goldbach's conjecture follows from this conjecture.
In this paper we introduce a simple method of searching for the prime pairs in the famous Goldbach Conjecture. The method, which is based on certain integer identities as well as an observation related to the remainder property, enables us to cut the search length by half and therefore can be considered as a short cut in searching for the prime pairs in the Goldbach Conjecture.
The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we further consider its applications to Goldbach's conjecture and refine the result in [11]. Moreover, we also try to generalize the problem of the least prime number in an arithmetic progression and give an analogy of Goldbach's conjecture.
The Goldbach conjecture that, every even integer is the sum of two primes, has been open since 1742. This paper details a road map to a proof of Goldbachs conjecture based on a function that estimates the number of Goldbach pairs. It is shown that the estimate of the number of Goldbach pairs increases as the even integer increases and that the error term is small. Further detailing the functions, assumptions and error term may finally lead to a proof of Goldbachs conjecture.
In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened form of the Chowla Hypothesis imply that every sufficiently large even integer may be written as the sum of two distinct primes. Résumé La conjecture de Goldbach et le plus petit nombre premier dans une progression arithmétique Dans ce document, nous essayons d'étudier les relations entre la conjecture de Goldbach et le plus petit nombre premier dans une progre…
We answer the question positively. In fact, we believe to have proved that every even integer $2N\geq3\times10^{6}$ is the sum of two odd distinct primes. Numerical calculations extend this result for $2N$ in the range $8-3\times10^{6}$. So, a fortiori, it is shown that every even integer $2N>2$ is the sum of two primes (Goldbach conjecture). Of course, we would be grateful for comments and objections.
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 paper presents some considerations about the Goldbach's conjecture (GC). The work is based on elementary results of the number theory and it provides a constructive method that permits, given an even integer, to find at least a pair of prime numbers according to the GC. Eventually a correlation will be provided between this constructive method and a class of problems of operations research.
The Strong Goldbach conjecture dates back to 1742. It states that every even integer greater than four can be written as the sum of two prime numbers. Since then, no one has been able to prove the conjecture. The only best known result so far is that every sufficiently large even integer N can be written as the sum of a prime number and the product of at most two prime numbers. Additionally, the conjecture has been verified to be true for all even integers up to 4.10^18. In this paper, we prove that the conjecture is true for all even integers greater than 362.
This article is a survey based on our earlier paper ("The 'Vertical' Generalization of the Binary Goldbach's Conjecture as Applied on 'Iterative' Primes with (Recursive) Prime Indexes (i-primeths)" [11]), a paper in which we have proposed a new generalization of the binary/"strong" Goldbach's Conjecture (GC) briefly called "the Vertical Goldbach's Conjecture" (VGC), which is essentially a meta-conjecture, as VGC states an in finite number of Goldbach-like conjectures stronger than GC, which all apply on "iterative" primes with recursive prime indexes (named "i-primes"). VGC was discovered by t…
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…
In this article we study in depth the Dirichlet theorem, which states that if a, b are relative prime integers, the sequence p = an + b contains infinite prime numbers, we simplify and generalize this theorem, we enunciate some special theorems (from Elias), we see equivalence with Bertrand's postulate and we deduce the validity of the Goldbach conjecture. Finally we enunciate some additional results that are deduced from the conjecture-theorem of Goldbach-
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…
Goldbach's comet is the plot of the Goldbach function g(n), in the interval [3, N], with a large positive integer N. The function g(n) counts the number of different ways in which 2n can be expressed as the sum of two odd primes. An account, hopefully satisfying and accessible, is given for the layers that make up the comet. By means of several (sometimes historical) results of Theory of Number, other conjectures, similar to the Goldbach's one, emerge. These are related with sequences of odd positive integers, like but not quite to the prime sequence.
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 …
The product m_k of the first k primes (2..p_k) has neighbours m_k +/- 1 with all prime divisors beyond p_k, implying there are infinitely many primes [Euclid]. All primes between p_k and m_k are in the group G_1 of units in semigroup Z_{m_k}(.) of mutiplication mod m_k. Due to the squarefree modulus Z_{m_k} is a disjoint union of 2^k groups, with as many idempotents - one per divisor of m_k, which form a Boolean lattice BL. The generators of Z_{m_k} and the additive properties of its lattice are studied. It is shown that each complementary pair in BL adds to 1 mod m_k and each even idempotent …
n 1937 Ivan Vinogradov proved the three prime sum version of the Goldbach Conjecture, often called the weak form of Goldbach Conjecture. And that it holds for "sufficiently large" odd natural numbers. In this work we use Dirichlet Theorem, Modulo Arithmetics, etc. to extended Vinogradov's Theorem such that every sufficiently large natural number (both even and odd) can be expressed as a sum of three primes. We highlight the configuration of primes for any special case of the three prime sum. Hence we obtain Vinogradov's Theorem as a special case of this extended version. We show how Vinogradov…
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 …
Mathematicians has been trying to prove the weak Goldbach's conjecture by adding prime numbers, as stated in the conjecture. However, we believe that the solution does not need to be analytically solved. Instead of trying to add prime numbers to prove the conjecture, we developed a prime number function P_{odd}(x)p>2, including odd primes p > 2, isomorphic and equivalent to a function N_{odd}(x)n>1, including odd natural numbers greater than one, n_{odd} > 1, in which, the sum of three of its elements result in odd numbers greater than 7, proving the conjecture.
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.
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 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…
We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cramér's, Shanks', and five Smarandache's conjectures. We make use of both Firoozbakht's conjecture (which recently proved by the author) and Kourbatov's theorem on the distribution of and gaps between consecutive primes. These latter conjecture and theorem play an essential role in our methods for proving these famous conjectures. In order to prove Shanks' conjecture, we make use of Panaitopol's asymptotic formula for $π(x)$ as well.
In this paper we introduce the prime index function \begin{align}ι(n)=(-1)^{π(n)},\nonumber \end{align} where $π(n)$ is the prime counting function. We study some elementary properties and theories associated with the partial sums of this function given by\begin{align}ξ(x):=\sum \limits_{n\leq x}ι(n).\nonumber \end{align}We show that a prime $p>2$ is a twin prime if and only if $ξ(p)=ξ(p+2)$. We also relate the prime index function to Cramer's conjecture by showing that \begin{align}|ξ(p_{n+1})-ξ(p_n)|+2=p_{n+1}-p_n.\nonumber \end{align}That is, Cramer's conjecture can be stated as \begin{alig…
In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime differences can be dealt with. Some, good results discussed here includes, the well know conjecture on prime gaps by Cramér and $\lim \inf_{n\to\infty} d_n < \infty$. Based on some simple assumptions, we have demonstrated how to tackle such problems.
