PreprintArticleVersion 3Preserved in Portico This version is not peer-reviewed
A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis. (Edition 8D)
Version 1
: Received: 11 September 2023 / Approved: 14 September 2023 / Online: 15 September 2023 (04:23:12 CEST)
Version 2
: Received: 21 September 2023 / Approved: 3 October 2023 / Online: 7 October 2023 (08:13:27 CEST)
Version 3
: Received: 17 November 2023 / Approved: 20 November 2023 / Online: 30 November 2023 (05:11:18 CET)
How to cite:
Buya, S.; Nchima, J.B. A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis. (Edition 8D). Preprints2023, 2023091012. https://doi.org/10.20944/preprints202309.1012.v3
Buya, S.; Nchima, J.B. A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis. (Edition 8D). Preprints 2023, 2023091012. https://doi.org/10.20944/preprints202309.1012.v3
Buya, S.; Nchima, J.B. A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis. (Edition 8D). Preprints2023, 2023091012. https://doi.org/10.20944/preprints202309.1012.v3
APA Style
Buya, S., & Nchima, J.B. (2023). A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis. (Edition 8D). Preprints. https://doi.org/10.20944/preprints202309.1012.v3
Chicago/Turabian Style
Buya, S. and John Bezaleel Nchima. 2023 "A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis. (Edition 8D)" Preprints. https://doi.org/10.20944/preprints202309.1012.v3
Abstract
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. The Riemann hypothesis is examined and sources of non trivial zeroes outside the critical strip are discussed. An example example of a non-trivial zero outside the critical strip is given
Keywords
proof of Binary Golbdach conjecture; proof of Andrica conjecture; proof of Legendre conjecture; Goldbach partition semiprime; disproof of Riemann hypothesis; proof of twin prime conjecture
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.
The commenter has declared there is no conflict of interests.
Comment:
Though this paper mainly deals with proofs of Binary Goldbach, Andrica and Legendre conjectures, it also deals with disproof of the Riemann hypothesis. Non trivial zeroes can be generated outside the critical strip the critical strip.
Example Result that contradicts the Riemann hypothesis
ζ(−1000 − i
1000π
log2
) = 0 (64)
This non-trivial zero is outside the critical strip. This result alone disproves the Riemann hypothesis.
Commenter: Samuel Buya
Commenter's Conflict of Interests: Author
Commenter:
The commenter has declared there is no conflict of interests.
Example Result that contradicts the Riemann hypothesis
ζ(−1000 − i
1000π
log2
) = 0 (64)
This non-trivial zero is outside the critical strip. This result alone disproves the Riemann hypothesis.