On Phi-Euler's Function in Refined Neutrosophic Number Theory and The Solutions of Fermat's Diophantine Equation

The objective of this paper is to answer the open problem proposed about the validity of phi-Euler’s theorem in the refined neutrosophic ring of integers (1, 2) . This work presents an algorithm to compute the values of Euler’s function on refined neutrosophic integers, and it prove that phi-Euler’s theorem is still true in (1, 2). On the other hand, we present a solution for another open question about the solutions of Fermat's Diophantine equation in refined neutrosophic ring of integers, where we determine the solutions of Fermat's Diophantine equation + = ; ≥ 3 in (1, 2)