A COMPARATIVE STUDY ON ADVANCEMENTS IN NEUTROSOPHIC NUMBER THEORETIC CONCEPTS IN THE NEUTROSOPHIC RING OF INTEGERS AND REFINED NEUTROSOPHIC RING OF INTEGERS

The study of neutrosophic numbers has garnered significant attention due to their ability to address uncertainty, indeterminacy, and contradictions in various mathematical models.Numbertheoretic concepts within the Neutrosophic Ring of Integers () are explored, including conditions for division, Euler's function, and congruences, along with other classical concepts.Additionally, further number-theoretic principles, such as division, congruences, and the existence of solutions to linear Diophantine equations, are examined and reviewed in the context of the Refined Neutrosophic Ring of Integers.This paper presents a comparative study between the neutrosophic ring of integers (NRI) and the refined neutrosophic ring of integers (RNRI).