Goldbach Conjecture (original) (raw)

Primes and Their Link to the Goldbach Conjecture

This informative paper, which is published in an international mathematics journal, presents insights and many important points on the prime numbers, which are the building-blocks or “atoms” of the integers, and the Goldbach conjecture formulated by Christian Goldbach (1690 - 1764) which are the result of years of research (the author having published two papers on the Goldbach conjecture in an international mathematics journal in 2012), 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 solved. The paper has much to share about the distribution or “behavior” of the prime numbers, providing much numerical evidence to support the conjecture, besides suggesting ways or arguments for resolving the conjecture.

A Generalization of Goldbach ’ S Conjecture

2017

Goldbach’s conjecture states that every even number greater than 2 can be expressed as the sum of two primes. The aim of this paper is to propose a generalization – or a set of increasingly generalized forms – of Goldbach’s conjecture and to present relevant computational results. The proposed statements also generalize Lemoine’s conjecture, according to which every sufficiently large odd integer is the sum of a prime and the double of another (or the same) prime. We present computational results verifying several cases of the statement until certain values and information regarding the resulting decompositions of even and odd integers.

Proof of Goldbach's Conjecture

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.

Some Considerations in Favor of the Truth of Goldbach’s Conjecture

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.

An elegant and short proof of Goldbach conjecture

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.