aspg-24-2724-1791482520 20261008180200 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Journal of Neutrosophic and Fuzzy Systems JNFS 2771-6430 2771-6449 2024 8 1 Warshine Barry University of Debrecen, Department of Mathematical and Computational Science, Debrecen, Hungary Lee Xu University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China Josef Al Jumayel Faculty Of Science, Beirut Arab University, Beirut, Lebanon The 3-cyclic refined neutrosophic roots of unity are exactly the solutions of the Diophantine equation style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;font-style:italic;mso-bidi-font-style:normal'> style='mso-bidi-font-style:normal'> "Cambria Math",serif;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font: major-bidi'>X normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi'>n style='mso-bidi-font-style:normal'> "Cambria Math",serif;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font: major-bidi'>=1 in the 3-cyclic refined neutrosophic ring of integers style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;font-style:italic;mso-bidi-font-style:normal'> style='mso-bidi-font-style:normal'> "Cambria Math",serif;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font: major-bidi'>Z normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi'>3 style='mso-bidi-font-style:normal'> "Cambria Math",serif;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font: major-bidi'>(I) . 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. 2024 23 30 2724 https://creativecommons.org/licenses/by/4.0/ 10.54216/JNFS.080103 https://www.americaspg.com/journal/24/article/2724 https://www.americaspg.com/storage/71714177688.pdf