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Considerations for Goldbach's Strong Conjecture
Version 1
: Received: 8 March 2021 / Approved: 10 March 2021 / Online: 10 March 2021 (13:14:22 CET)
Version 2 : Received: 12 March 2021 / Approved: 12 March 2021 / Online: 12 March 2021 (11:29:49 CET)
Version 2 : Received: 12 March 2021 / Approved: 12 March 2021 / Online: 12 March 2021 (11:29:49 CET)
How to cite: Severino Silva, C. Considerations for Goldbach's Strong Conjecture. Preprints 2021, 2021030281. https://doi.org/10.20944/preprints202103.0281.v2 Severino Silva, C. Considerations for Goldbach's Strong Conjecture. Preprints 2021, 2021030281. https://doi.org/10.20944/preprints202103.0281.v2
Abstract
Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.
Keywords
Goldbach's conjecture; numbers prime; Arithmetic Theorem
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Commenter: Carleilton Severino Silva
Commenter's Conflict of Interests: Author
This article will be limited to only a few properties of the whole numbers that are deemed necessary for the logical construction of this work. The Set of Integers is represented by ℤ and the number of elements contained in ℤ is infinite, that is, ℤ ={…,−4,−3,−2,−1,0,1,2,3,4,…}. ℤ can be represented without the neutral element ℤ∗={…,−4,−3,−2,−1,1,2,3,4,…}. The Set of Natural Numbers