On The Group of Units Classification In 3-Cyclic and 4-cyclic Refined Rings of Integers And The Proof of Von Shtawzens' Conjectures

Hasan; Mohammad

doi:https://doi.org/10.54216/IJNS.210414

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Volume 21 • Issue 4 • PP: 146-154 • 2023

On The Group of Units Classification In 3-Cyclic and 4-cyclic Refined Rings of Integers And The Proof of Von Shtawzens' Conjectures

Hasan Sankari ^1*

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,

Mohammad Abobala ¹

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¹Tishreen University, Department Of Mathematics, Latakia, Syria

* Corresponding Author.

DOI https://doi.org/10.54216/IJNS.210414

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Received: January 06, 2023 Revised: March 12, 2023 Accepted: April 28, 2023

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Abstract

First Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with three variables . This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 6. Second Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with four variables. This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 8. In this paper, we prove that first Von Shtawzen's conjecture is true, where we show that first Von Shtawzen's Diophantine equations has exactly 12 solutions. On the other hand, we find all solutions of this Diophantine equations. In addition, we provide a full proof of second Von Shtawzen's conjecture, where we prove that the previous Diophantine equation has exactly 16 solutions, and we determine all of its possible solutions

Keywords

n-cyclic refined ring first Von Shtawzen's conjecture group of units second Von Shtawzen's conjecture

References

[1] D. G. Northcott, Lessons on Rings, Modules, and Multiplicities, Cambridge Univ. Press, Cambridge, 1968.

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

[3] 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.

[4] 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.

[5] 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.

[6] 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.

[7] 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.

[8] Sarkis, M., " On The Solutions Of Fermat's Diophantine Equation In 3-refined Neutrosophic Ring of Integers", Neoma Journal of Mathematics and Computer Science, 2023.

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Sankari, Hasan, Abobala, Mohammad. "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, vol. Volume 21, no. Issue 4, 2023, pp. 146-154. DOI: https://doi.org/10.54216/IJNS.210414

Sankari, H., Abobala, M. (2023). 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, Volume 21(Issue 4), 146-154. DOI: https://doi.org/10.54216/IJNS.210414

Sankari, Hasan, Abobala, Mohammad. "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 Volume 21, no. Issue 4 (2023): 146-154. DOI: https://doi.org/10.54216/IJNS.210414

Sankari, H., Abobala, M. (2023) '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, Volume 21(Issue 4), pp. 146-154. DOI: https://doi.org/10.54216/IJNS.210414

Sankari H, 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;Volume 21(Issue 4):146-154. DOI: https://doi.org/10.54216/IJNS.210414

H. Sankari, M. Abobala, "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, vol. Volume 21, no. Issue 4, pp. 146-154, 2023. DOI: https://doi.org/10.54216/IJNS.210414

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