International Journal of Neutrosophic Science

Journal DOI

https://doi.org/10.54216/IJNS

Submit Your Paper

2690-6805ISSN (Online) 2692-6148ISSN (Print)

Volume 20 , Issue 3 , PP: 08-14, 2023 | Cite this article as | XML | Html | PDF | Full Length Article

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.

    [6] Ceven, Y., and Tekin, S., " Some Properties of Neutrosophic Integers", Kırklareli University Journal of Engineering and Science, Vol. 6, 2020.

    [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 :
    M., Abuobida. , A., Yaser. , A., Raja. , Ali, Rozina. On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. International Journal of Neutrosophic Science, vol. , no. , 2023, pp. 08-14. DOI: https://doi.org/10.54216/IJNS.200301
    M., A. A., Y. A., R. Ali, R. (2023). On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. International Journal of Neutrosophic Science, (), 08-14. DOI: https://doi.org/10.54216/IJNS.200301
    M., Abuobida. A., Yaser. A., Raja. Ali, Rozina. On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. International Journal of Neutrosophic Science , no. (2023): 08-14. DOI: https://doi.org/10.54216/IJNS.200301
    M., A. , A., Y. , A., R. , Ali, R. (2023) . On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. International Journal of Neutrosophic Science , () , 08-14 . DOI: https://doi.org/10.54216/IJNS.200301
    M. A. , A. Y. , A. R. , Ali R. [2023]. On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers. International Journal of Neutrosophic Science. (): 08-14. DOI: https://doi.org/10.54216/IJNS.200301
    M., A. A., Y. A., R. Ali, R. "On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers," International Journal of Neutrosophic Science, vol. , no. , pp. 08-14, 2023. DOI: https://doi.org/10.54216/IJNS.200301