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International Journal of Neutrosophic Science
Volume 20 , Issue 3, PP: 08-14 , 2023 | Cite this article as | XML | Html |PDF

Title

On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers

  Abuobida M. A. Alfahal 1 * ,   Yaser A. Alhasan 2 ,   Raja A. Abdulfatah 3 ,   Rozina Ali 4

1  Deanship of the Preparatory Year, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia
    (a.alfahal@psau.edu.sa)

2  Deanship of the Preparatory Year, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia
    (y.alhasan@psau.edu.sa)

3  Deanship of the Preparatory Year, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia
    (r.abdulfatah@psau.edu.sa)

4  Cairo University, department of mathematics, Cairo , Egypt
    (Rozyyy123n@gmail.com)


Doi   :   https://doi.org/10.54216/IJNS.200301

Received: August 20, 2022 Accepted: March 02, 2023

Abstract :

The Diophantine equation X^n+Y^n=Z^n is called the Fermat's Diophantine equation. Its solutions are called general Fermat's triples.The aim of this paper is to study the solutions of Fermat's Diophantine equation in the 2-cyclic refined neutrosophic ring of integers, where we determine all possible solutions of this Diophantine equation, as well as, the special case of Pythagoras triples.

Keywords :

2- cyclic refined Neutrosophic integer; 2-cyclic refined Neutrosophic Fermat's equation; Pythagoras triple 

References :

[1] Smarandache, F., " A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability", American Research Press. Rehoboth, 2003.

[2] Sankari, H., and Abobala, M., "Neutrosophic Linear Diophantine Equations With two Variables", Neutrosophic Sets and Systems, Vol. 38, pp. 22-30, 2020.

[3] Adeleke, E.O., Agboola, A.A.A.,and Smarandache, F., "Refined Neutrosophic Rings I", International Journal of Neutrosophic Science, Vol. 2(2), pp. 77-81, 2020.

[4] Ali, R., "A Short Note On The Solution of n-Refined Neutrosophic Linear Diophantine Equations", International Journal Of Neutrosophic Science, Vol. 15, 2021.

[5] Ibrahim, M., and Abobala, M., "An Introduction To Refined Neutrosophic Number Theory", Neutrosophic Sets and Systems, Vol. 45, 2021.

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[7] Ahmad, K., Bal, M., and Aswad, M., " A Short Note On The Solutions Of Fermat's Diophantine Equation In Some Neutrosophic Rings", Journal Of Neutrosophic and Fuzzy Systems, 2022.

[8] Abobala, M., Partial Foundation of Neutrosophic Number Theory, Neutrosophic Sets and Systems, Vol. 39 , 2021.

[9] Celik, M., and Olgun, N., " An Introduction To Neutrosophic Real Banach And Hillbert Spaces", Galoitica Journal Of Mathematical Structures And Applications, 2022.

[10] Celik, M., and Olgun, N., " On The Classification Of Neutrosophic Complex Inner Product Spaces", Galoitica Journal Of Mathematical Structures And Applications, 2022.

[11] Celik, M., and Hatip, A., " On The Refined AH-Isometry And Its Applications In Refined Neutrosophic Surfaces", Galoitica Journal Of Mathematical Structures And Applications, 2022

[12] Smarandache, F., "Symbolic Neutrosophic Theory", EuropaNova asbl, Bruxelles, 2015.

[13] Merkepci, H., and Ahmad, K., " On The Conditions Of Imperfect Neutrosophic Duplets and Imperfect Neutrosophic Triplets", Galoitica Journal Of Mathematical Structures And Applications, Vol.2, 2022.

[14] Merkepci, H., and Abobala, M., " The Application of AH-isometry In The Study Of Neutrosophic Conic Sections", Galoitica Journal Of Mathematical Structures And Applications, Vol.2, 2022.

[15] Abobala, M., and Zeina, M.B., " A Study Of Neutrosophic Real Analysis By Using One Dimensional Geometric AH-Isometry", Galoitica Journal Of Mathematical Structures And Applications, Vol.3, 2023

[16] Hatip, A., "An Introduction To Weak Fuzzy Complex Numbers ", Galoitica Journal Of Mathematical Structures and Applications, Vol.3, 2023

[17] Kandasamy, V.W.B., and Smarandache, F., "Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures", Hexis, Phonex, Arizona 2006.

[18] Smarandache, F., n-Valued Refined Neutrosophic Logic and Its Applications in Physics, Progress in Physics, 143-146, Vol. 4, 2013.

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


Cite this Article as :
Style #
MLA Abuobida M. A. Alfahal, Yaser A. Alhasan, Raja A. Abdulfatah, Rozina Ali. "On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers." International Journal of Neutrosophic Science, Vol. 20, No. 3, 2023 ,PP. 08-14 (Doi   :  https://doi.org/10.54216/IJNS.200301)
APA Abuobida M. A. Alfahal, Yaser A. Alhasan, Raja A. Abdulfatah, Rozina Ali. (2023). On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. Journal of International Journal of Neutrosophic Science, 20 ( 3 ), 08-14 (Doi   :  https://doi.org/10.54216/IJNS.200301)
Chicago Abuobida M. A. Alfahal, Yaser A. Alhasan, Raja A. Abdulfatah, Rozina Ali. "On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers." Journal of International Journal of Neutrosophic Science, 20 no. 3 (2023): 08-14 (Doi   :  https://doi.org/10.54216/IJNS.200301)
Harvard Abuobida M. A. Alfahal, Yaser A. Alhasan, Raja A. Abdulfatah, Rozina Ali. (2023). On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. Journal of International Journal of Neutrosophic Science, 20 ( 3 ), 08-14 (Doi   :  https://doi.org/10.54216/IJNS.200301)
Vancouver Abuobida M. A. Alfahal, Yaser A. Alhasan, Raja A. Abdulfatah, Rozina Ali. On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. Journal of International Journal of Neutrosophic Science, (2023); 20 ( 3 ): 08-14 (Doi   :  https://doi.org/10.54216/IJNS.200301)
IEEE Abuobida M. A. Alfahal, Yaser A. Alhasan, Raja A. Abdulfatah, Rozina Ali, On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers, Journal of International Journal of Neutrosophic Science, Vol. 20 , No. 3 , (2023) : 08-14 (Doi   :  https://doi.org/10.54216/IJNS.200301)