Goldbach Conjecture

International journal of pure and applied mathematics, 2012

The author had published a 14-page paper on the solutions for the twin primes conjecture in the International Mathematical Journal in 2003. This paper, which comprises of 3 parts that are each self-contained, presents various approaches to the Goldbach conjecture, which is a related problem.

2023

In this paper, we present a direct proof of the Goldbach Conjecture for even numbers greater than 2, a longstanding problem in number theory. Our proof builds upon known results and well-established principles of elementary number theory, including the weak Goldbach conjecture and properties of even and odd numbers, ultimately establishing the validity of the conjecture, through rigorous mathematical reasoning and deductive approach. The Goldbach Conjecture posits that every even number greater than 2 can be expressed as the sum of two prime numbers. We provide a clear and concise proof that reaffirms the conjecture's accuracy, contributing to the body of knowledge in prime number theory and tries to prove one of the most famous unsolved problems in number theory.

When considering whether every even integer can be expressed as the sum of two primes, it is tempting to view the puzzle as a question of arithmetic, while the answer lies in the infinite pattern of the primes. Instead of attempting to prove that every even integer has this property, and ignoring the noticeable pattern that larger even numbers have more prime pairs, we attempt to find an even number without this property, consider how many of these such numbers can exist, and come to the conclusion that no such number can exist. Because it is simple to show that small numbers (no more than three digits) have the property, we are most concerned with larger numbers, considering whether the widening gaps between larger primes may eventually create an even number that cannot be expressed between the sum of the plentiful small primes and the highly scarce large primes. In our exploration, we will mathematically create the ideal scenario for a non-prime-sum number to exist, then show that as this scenario extends towards the infinite, a non-prime-sum number does not become more possible, but in fact less possible. We will additionally see that extending toward the infinite, the number of prime pairs that sum to a given even number will increase without limit.

2024

This paper provides an elementary proof of Strong Goldbach's conjecture stating that every even natural number greater than two can be written as the sum of two primes. The conjecture addresses one of the oldest and most fundamental problems in number theory, which is understanding the distribution and properties of prime numbers. The proof employs a structural approach to numbers. On one hand, it builds upon the understanding of the structure of primes, specifically focusing on their 6k ± 1 patterns (𝑝𝑟𝑖𝑚𝑒𝑠 ≥ 5) as discussed in the opening section of the document, and on the other hand, it explores the underlying structure of even numbers, classifying them into three distinct categories based on their relationship to multiples of 3. Additionally, the proof employs a sieve logic encapsulated within an ad-hoc notation introduced in this paper (ad-hoc sieve notation), to provide the description for prime numbers into which an even number, according to its structure, can be decomposed. Finally, the proof covers formally the even numbers greater or equal to 10. The even numbers 4,6 and 8 are cases accepted as outliers in this approach since they involve the particular primes 2 and 3 that are not fitting any of 6k ± 1 prime patterns (𝑎𝑠: 4 = 2 + 2, 6 = 3 + 3 𝑎𝑛𝑑 8 = 3 + 5). At the conclusion of this paper, several practical and direct implications of this approach are discussed. These implications involve the identification of twin primes, new primality testing, identifying sequences of primes, and the articulation with the Goldbach's Comet.

Example result that contradicts the Riemann hypothesis. A further examination of gaps between two consecutive primes 15 Arithmetic mean and geometric mean perspectives in prime gaps. .

This article presents some considerations about the Goldbach’s conjecture (GC). The work is based on analytic 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. It will be shown how the method can be implemented by an algorithm coded in a high-level language for numerical computation. Eventually a correlation will be provided between this constructive method and a class of problems of operations research.

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 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.

In this research a neccessary and sufficient condition for the proof of the Binary Goldbach conjecture is established. It is established that the square of all natural numbers greater or equal to 2 have an additive partition equal to the sum of the square of a natural number greater or equal to zero and a Goldbach partition semiprime. All Goldbach partition semiprimes are odd except 4. This finding is in itself proof all composite even numbers have at least Goldbach partition. The result of the proof of the Binary Goldbach conjecture is used to prove the Andrica and Legendre conjectures.

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.