On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu; Jetmira Uka

doi:10.20944/preprints202110.0087.v17

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preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202110.0087.v17

Preprint Article Version 17 Preserved in Portico This version is not peer-reviewed

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu

^* and

Jetmira Uka

Version 1 : Received: 1 October 2021 / Approved: 5 October 2021 / Online: 5 October 2021 (14:48:18 CEST)
Version 2 : Received: 24 November 2021 / Approved: 25 November 2021 / Online: 25 November 2021 (20:28:09 CET)
Version 3 : Received: 4 December 2021 / Approved: 7 December 2021 / Online: 7 December 2021 (20:52:47 CET)
Version 4 : Received: 12 January 2022 / Approved: 14 January 2022 / Online: 14 January 2022 (14:43:54 CET)
Version 5 : Received: 29 June 2022 / Approved: 30 June 2022 / Online: 30 June 2022 (03:50:17 CEST)
Version 6 : Received: 9 July 2022 / Approved: 11 July 2022 / Online: 11 July 2022 (04:28:25 CEST)
Version 7 : Received: 21 July 2022 / Approved: 22 July 2022 / Online: 22 July 2022 (03:38:39 CEST)
Version 8 : Received: 4 August 2022 / Approved: 5 August 2022 / Online: 5 August 2022 (03:41:27 CEST)
Version 9 : Received: 17 August 2022 / Approved: 18 August 2022 / Online: 18 August 2022 (03:53:01 CEST)
Version 10 : Received: 27 August 2022 / Approved: 30 August 2022 / Online: 30 August 2022 (03:11:25 CEST)
Version 11 : Received: 12 September 2022 / Approved: 13 September 2022 / Online: 13 September 2022 (10:15:57 CEST)
Version 12 : Received: 26 September 2022 / Approved: 27 September 2022 / Online: 27 September 2022 (03:49:33 CEST)
Version 13 : Received: 10 October 2022 / Approved: 11 October 2022 / Online: 11 October 2022 (03:22:25 CEST)
Version 14 : Received: 14 November 2022 / Approved: 15 November 2022 / Online: 15 November 2022 (06:35:57 CET)
Version 15 : Received: 21 November 2022 / Approved: 22 November 2022 / Online: 22 November 2022 (03:01:41 CET)
Version 16 : Received: 12 December 2022 / Approved: 13 December 2022 / Online: 13 December 2022 (02:17:30 CET)
Version 17 : Received: 29 December 2022 / Approved: 30 December 2022 / Online: 30 December 2022 (03:45:58 CET)
Version 18 : Received: 10 July 2023 / Approved: 12 July 2023 / Online: 13 July 2023 (10:01:00 CEST)
Version 19 : Received: 8 August 2023 / Approved: 9 August 2023 / Online: 10 August 2023 (05:23:00 CEST)
Version 20 : Received: 8 February 2024 / Approved: 9 February 2024 / Online: 9 February 2024 (11:19:03 CET)
Version 21 : Received: 10 February 2024 / Approved: 12 February 2024 / Online: 13 February 2024 (07:15:16 CET)
Version 22 : Received: 3 April 2024 / Approved: 3 April 2024 / Online: 4 April 2024 (17:39:40 CEST)

How to cite: Shehu, A.; Uka, J. On the Additive and Subtractive Representation of Even Numbers from Primes. Preprints 2021, 2021100087. https://doi.org/10.20944/preprints202110.0087.v17 Shehu, A.; Uka, J. On the Additive and Subtractive Representation of Even Numbers from Primes. Preprints 2021, 2021100087. https://doi.org/10.20944/preprints202110.0087.v17

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MDPI and ACS Style

Shehu, A.; Uka, J. On the Additive and Subtractive Representation of Even Numbers from Primes. Preprints 2021, 2021100087. https://doi.org/10.20944/preprints202110.0087.v17

APA Style

Shehu, A., & Uka, J. (2022). On the Additive and Subtractive Representation of Even Numbers from Primes. Preprints. https://doi.org/10.20944/preprints202110.0087.v17

Chicago/Turabian Style

Shehu, A. and Jetmira Uka. 2022 "On the Additive and Subtractive Representation of Even Numbers from Primes" Preprints. https://doi.org/10.20944/preprints202110.0087.v17

Abstract

We demonstrate a new quantitative method to the sieve of Eratosthenes, which is an alternative to the sieve of Legendre. In this method, every element of a given set is sifted out once only, and therefore, this method is free of the Mobius function and of the parity barrier. Using this method, we prove that every sufficiently large even number is the sum of two primes, and that every even number is the difference of two primes in infinitely many ways.

Keywords

Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; 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.

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Comments (1)

Comment 1

Received: 30 December 2022
Commenter: Ali Shehu
Commenter's Conflict of Interests: Author

Comment: A minor clarifications.

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