On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity

Warshine; Lee; Josef Al

doi:https://doi.org/10.54216/JNFS.080103

Full Length Article

Volume 8 • Issue 1 • PP: 23-30 • 2024

On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity

Warshine Barry ^1*

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,

Lee Xu ²

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,

Josef Al Jumayel ³

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¹University of Debrecen, Department of Mathematical and Computational Science, Debrecen, Hungary

²University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China

³Faculty Of Science, Beirut Arab University, Beirut, Lebanon

* Corresponding Author.

DOI https://doi.org/10.54216/JNFS.080103

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Received: October 17, 2023 Revised: December 16, 2023 Accepted: January 24, 2024

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Abstract

The 3-cyclic refined neutrosophic roots of unity are exactly the solutions of the Diophantine equation in the 3-cyclic refined neutrosophic ring of integers . This paper is dedicated to finding all 3-cyclic refined neutrosophic Diophantine roots of unity, where it proves that there exist only three solutions for the case of odd order (n), and twelve different solutions for the case of even order (n). On the other hand, the group generated from all solutions will be classified as a finite abelian group with direct products of finite cyclic groups.

Keywords

3-cyclic refined neutrosophic rings 3-cyclic refined Diophantine root of unity Von Shtawzen's group of units neutrosophic unit

References

[1] Basheer, A., Ahmad, K., and Ali, R., " A Short Contribution To Von Shtawzen's Abelian Group In n-Cyclic Refined Neutrosophic Rings", Journal Of Neutrosophic And Fuzzy Systems,, 2022.

[2] Von Shtawzen, O., " Conjectures For Invertible Diophantine Equations Of 3-Cyclic and 4-Cyclic Refined Integers", Journal Of Neutrosophic And Fuzzy Systems, Vol.3, 2022.

[3] Von Shtawzen, O., " On A Novel Group Derived From A Generalization Of Integer Exponents and

Open Problems", Galoitica journal Of Mathematical Structures and Applications, Vol 1, 2022.

[4] Basheer, A., Ahmad, K., and Ali, R., " On Some Open Problems About n-Cyclic Refined Neutrosophic Rings and Number Theory", Journal Of Neutrosophic And Fuzzy Systems,, 2022.

[5] A. Alrida Basheer , Katy D. Ahmad , Rozina Ali., "Examples on Some Novel Diophantine Equations Derived from the Group of Units Problem in n-Cyclic Refined Neutrosophic Rings of Integers", Galoitica Journal Of Mathematical Structures And Applications, Vol.3, 2022.

[6] Sankari, H., and Abobala, M., " On The Group of Units Classification In 3-Cyclic and 4-cyclic Refined Rings of Integers And The Proof of Von Shtawzens' Conjectures", International Journal of Neutrosophic Science, 2023.

[7] Sankari, H., and Abobala, M., " On The Classification of The Group of Units of Rational and Real 2-Cyclic Refined Neutrosophic Rings", Neutrosophic Sets and Systems, 2023.

[8] Sankari, H., and Abobala, M., " On The Algebraic Homomorphisms Between Symbolic 2-Plithogenic Rings And 2-cyclic Refined Rings", Neutrosophic Sets and Systems, 2023.

[9] Abobala, M., " n-Cyclic Refined Neutrosophic Algebraic Systems of Sub-Indeterminacies, An Application to Rings and Modules", International Journal of Neutrosophic Science, 2020.

[10] Aswad, M., "n-Cyclic Refined Neutrosophic Vector Spaces and Matrices", Neutrosophic Knowledge, 2021.

[11] Ali, R., "n-Cyclic Refined Neutrosophic Groups", International Journal of Neutrosophic Science, 2021.

[12] Sadiq. B., " A Contribution To The group Of Units Problem In Some 2-Cyclic Refined Neutrosophic Rings ", International Journal Of Neutrosophic Science, 2022.

[13] Nabil Khuder Salman, Maikel Leyva Vazquez,Batista Hernández Noel. On The Classification of 3-Cyclic/4-Cyclic Refined Neutrosophic Real and Rational Von Shtawzen's Group. International Journal of Neutrosophic Science, (2024); 23 ( 2 ): 26-31.

[14] Ahmad Salama, Rasha Dalla, Malath Al Aswad, Rozina Ali. (2022). Some Results About 2-Cyclic Refined Neutrosophic Complex Numbers. Journal of Neutrosophic and Fuzzy Systems, 4 ( 1 ), 41-8 (Doi : https://doi.org/10.54216/JNFS.040105).

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Barry, Warshine, Xu, Lee, Jumayel, Josef Al. "On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity." Journal of Neutrosophic and Fuzzy Systems, vol. Volume 8, no. Issue 1, 2024, pp. 23-30. DOI: https://doi.org/10.54216/JNFS.080103

Barry, W., Xu, L., Jumayel, J. (2024). On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity. Journal of Neutrosophic and Fuzzy Systems, Volume 8(Issue 1), 23-30. DOI: https://doi.org/10.54216/JNFS.080103

Barry, Warshine, Xu, Lee, Jumayel, Josef Al. "On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity." Journal of Neutrosophic and Fuzzy Systems Volume 8, no. Issue 1 (2024): 23-30. DOI: https://doi.org/10.54216/JNFS.080103

Barry, W., Xu, L., Jumayel, J. (2024) 'On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity', Journal of Neutrosophic and Fuzzy Systems, Volume 8(Issue 1), pp. 23-30. DOI: https://doi.org/10.54216/JNFS.080103

Barry W, Xu L, Jumayel J. On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity. Journal of Neutrosophic and Fuzzy Systems. 2024;Volume 8(Issue 1):23-30. DOI: https://doi.org/10.54216/JNFS.080103

W. Barry, L. Xu, J. Jumayel, "On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity," Journal of Neutrosophic and Fuzzy Systems, vol. Volume 8, no. Issue 1, pp. 23-30, 2024. DOI: https://doi.org/10.54216/JNFS.080103

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