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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)
Buya, S.B.; 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. International Journal of Pure and Applied Mathematics Research 2024, 4, 12–27, doi:10.51483/ijpamr.4.1.2024.12-27.
Buya, S.B.; 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. International Journal of Pure and Applied Mathematics Research 2024, 4, 12–27, doi:10.51483/ijpamr.4.1.2024.12-27.
Buya, S.B.; 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. International Journal of Pure and Applied Mathematics Research 2024, 4, 12–27, doi:10.51483/ijpamr.4.1.2024.12-27.
Buya, S.B.; 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. International Journal of Pure and Applied Mathematics Research 2024, 4, 12–27, doi:10.51483/ijpamr.4.1.2024.12-27.
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
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.