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

Abstract

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 𝑍(𝐼1,𝐼2) . 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 𝑍(𝐼1,𝐼2).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 𝑋𝑛 + 𝑌𝑛 = 𝑍𝑛; 𝑛 ≥ 3 in 𝑍(𝐼1,𝐼2). Key Words: refined Neutrosophic integer, Neutrosophic Euler's function, Neutrosophic, Fermat's equation