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.
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