Partial Foundation of Neutrosophic Number Theory

Abstract

The aim of this paper is to establish a partial foundation of number-theoretical concepts in the neutrosophic ring of integers Z(I) because it is based on a partial order relationship. This work partially generalizes and deals with necessary and sufficient conditions for division, Euler's function, congruencies, and some other classical concepts in Z(I). The main result of this work is to show that Euler's famous theorem is still true in the case of neutrosophic integers for our partial ordering relationship. Also, this work introduces an algorithm to solve Pell's equation in the neutrosophic ring of integers Z(I).