Goldbach Conjecture Proof

https://www.academicjournals.org/journal/AJMCSR/edition/August_2018, 2018

In this paper several methods are examined for proving the Goldbach conjecture. At the preliminary analysis stage a Diophantine equation solution method is proposed for Goldbach partition of a Goldbach number. The proof method proposed however is found to be incomplete since it does not have mechanisms for dealing with the prime gap problem. On the further analysis section some graphical and linear analytical methods are proposed for Goldbach partition as an extension of the solution of proposed quadratic equation. The Riemann hypothesis is examined in light of some findings on Goldbach conjecture. A proof is then proposed for the Riemann hypothesis. The proof results are used to attempt to prove Goldbach conjecture but without success. A justification for proof by induction method is proposed. A theorem 1 is proposed by an attempt is made to prove the conjecture by induction. To reinforce the proof by induction, a Samuel –Goldbach theorem is proved in which it is shown that any even number greater than six is the sum of four prime numbers. The theorem is then reduced to Goldbach strong and weak conjectures. Goldbach weak conjecture (proved) is also reduced to the strong conjecture. A proof method is thus proposed by which the weak conjecture is reduced to the strong. The proof method however is not completely satisfactory because it does not provide an analytical solution of the prime gap problem. Proof method however gave lead to the importance of even numbers in Goldbach partition. A proof method of proving the Goldbach conjecture is discussed by which each odd prime number is connected to a specific even number. Through this connection a family of curves with even number points for Goldbach partition of a Goldbach number is proposed. The family of curves containing these special even coordinate points helps overcome the prime gap problem in Goldbach partition. It is found that each Goldbach number has at least one pair of these special even numbers to enable Goldbach partition. A special identity then used to come up with a special quadratic function for Goldbach partition. The function has at least one point with an x coordinate representing gap between primes of the Goldbach partition any a y coordinate that is a product of the same primes. Thus Golbach conjecture is fully proved and the prime gap problem of the partition solved.

The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture referred to as "all even numbers that is greater than 2 can be expressed as the sum-two primes). Yet, no proof of Goldbach's Conjecture has been found. This assumption seems to be right for a large multitude of numbers using numerical calculations. Some examples are 10 = 3 + 7, 18 = 7 + 11, 100 = 97 + 3, and so on. But there is a multitude of primes of even numbers which, i would say that it is irregular, but increases with the order of the even numbers. The same happens and with an odd number as example 25, ie. 25 = 3 + 3 + 19 = 3 + 5 + 17 = 3 + 11 + 11 = 5 + 7 + 13 = 7 + 7 + 11. We say only 5 cases and only those that meet the constant sum of 25. First turns in Theorem 3 that there may be at least a pair of primes, such that the sum is equal to every even number. But at the same time reveals that the method of finding all pairs satisfy this condition. According to Theorem 4, the second guess interrupted to form the first guess and this is primarily elementary, to prove the truth of 1. The alternative and the side stream assistant, for the prove Goldbach's Conjecture in this research, was the Mathematica program which is the main tool for data collection, but also for finding all the steps in order to demonstrate each part of the proof. We must mention that Vinogradov is proving to 1937, that for every sufficiently large number can be expressed, that is the sum of the three primes. And finally, the Chinese mathematician Chen Jing as demonstrated for a big prime and with a constant number that is the sum of the tree first in 1966 [2]. Finally, the investigation of Goldbach's Conjecture has acted as a catalyst for the creation and development of many methods that are useful, with many theorems that help and other areas of mathematics.

Mantzakouras Nikos(10.13140 / RG.2.2.32893.69600), 2013

Every even integer > 2 is the sum of two prime numbers & equivalent Each odd integer > 5 is the sum of three prime numbers USING THE SIEVE OF ERATOSTHENES

ResearchGate, 2024

When analysing the Goldbach Conjecture it is useful to partition the Goldbach Sum, 2g, into two types of sets: those where 2g = 6k, and those where 2g = 6k±2. Primes greater than 3 can be classified as either 6k+1 or 6k-1. If we model the Goldbach conjecture using an equal density of 6k+1 and 6k-1 primes, and further model the upper primes Q as being randomly distributed with respect to the lower primes P, with each odd summand pair independent from the others, we get an excellent estimate for the Goldbach Count, and this empirically validates the model.

Vixra

In this research a general algebraic relationship is established between a pair of primes. This relationship shown to be a useful tool to prove the Binary Goldbach conjecture and other conjectures involving prime gap. A partition approach without parity obstruction will be presented. Through the approach it is shown that every composite even number has at least one Goldbach partition. Keywords Algebraic relationship between primes; proof of binary Goldbach conjecture; Quadratic equation for solving the prime gap problem; A partition approach without parity obstruction; equation establishing interconnectivity between primes;

Goldbach's famous conjecture has always fascinated eminent mathematicians. In this paper we give a rigorous proof basedon a new formulation, namely, that every even integer has a primo-raduis. Our proof is mainly based on the application ofChebotarev-Artin's theorem, Mertens' formula and the Principle exclusion-inclusion of Moivre.

This paper is a revision and expansion of two papers on the Goldbach conjecture which the author had published in an international mathematics journal in 2012. It presents insights and many important points on the conjecture and the prime numbers which are the result of years of research, all of which would be of interest to researchers working on the prime numbers and the Goldbach conjecture itself. The Goldbach conjecture, viz., every even number after 2 is the sum of 2 primes, is actually related to the distribution or "behavior" of the prime numbers. Therefore, when the distribution or "behavior" of the prime numbers is firmly understood the conjecture could be more easily resolved. This paper, which has been refereed and accepted for publication, has much to share about the distribution or "behavior" of the prime numbers, besides resolving the conjecture.

2018

The aim of this study is to prove Goldbach's famous Conjectures known as strong and weak conjectures. Strong Conjecture: "Every even number greater than 2 is the sum of two prime numbers". Weak Conjecture: "Every odd integer greater than 5 can be written as the sum of three prime numbers". Content: Searching prime numbers with predictive formulas is beyond the scope of this work. The program that generates and tests the prime number is a separate study, in this study the proofs of the Strong and Weak Goldbach Conjectures will take place. We only approach Gold-bach's Conjecture, where all required prior knowledge on prime numbers assumed as accepted by Goldbach's works. Therefore, we start with take a look at Goldbach's description of original problem then will try to derive a step by step proof upon that is described as in original letters. Method: Once the sub groups of the set of prime numbers were defined, theoretical framework was proved to be complete. The theoretical framework is very simple and concise, albeit the entire study is based upon that. This study is providing proof on both of conjectures. Findings and Results: In this study, an effective solution to a historical problem known in mathematics but not proven to this day is introduced; the proofs of Goldbach's Conjectures (both in Strong and Weak) are given. These proofs will open many obstacles in number theory and provide a fresh look on prime numbers and their applications. Many assumptions, conjectures in number theory will be re-evaluated. Based on this proof, another theorem on prime number is constructed with its proof. Conclusion: Many assumptions, conjectures about prime numbers will be re-evaluated under the light of given proofs. There is no reason to limit the sum in the theorem above; one can easily say that prime numbers are infinite. As a discussion, albeit there are institutionalized methods on computing prime numbers, a new way of computing bigger prime numbers can be based on this new perspective this paper has shed lights on. Proofs presented will introduce new horizons to relevant academicians on number theory. Defining this new perspective might also help one to expend this study further points related to even if there is a pattern on Prime Numbers so we can exploit that to compute bigger numbers feasibly. The prominence of major prime numbers in cryp-tology is known. Encryption will be redesigned in line with the proofs. The proof will open new horizons on prime numbers, the first one is this explained and proved new Prime Number Theorem: Aksoy Theorem.

The Goldbach conjecture dates back to 1742 ; we refer the reader to [1]-[2] for a history of the conjecture. Christian Goldbach stated that every odd integer greater than seven can be written as the sum of at most three prime numbers. Leonhard Euler then made a stronger conjecture that every even integer greater than four can be written as the sum of two primes. Since then, no one has been able to prove the Strong Goldbach Conjecture. The only best known result so far is that of Chen [3], proving that every suciently 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 veried to be true for all even integers up to 4.10 18 in 2014 , Jërg [4] and Tomás [5]. In this paper, we prove that the conjecture is true for all even integers greater than 8.

We will prove that the number of prime pairs, i.e., the representations of an even integer x as a sum of two primes, is almost the same result as that conjectured by Hardy and Littlewood [1]. First, a sliding model is proposed and applied to preserve all of the prime pairs without adopting a probabilistic estimation leading to an uncertain result. Next, the strong and weak even patterns are defined and used to determine the ability to gain prime pairs of different types of evens. Finally, we conclude that Goldbach's conjecture is true.