Foundations of Neutrosophic Number Theory

The aim of this paper is to establish a strong foundation of number theoretical concepts in the neutrosophic ring of integers 𝑍(𝐼). This work 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. Also, this work introduces an algorithm to solve Pell's equation in the neutrosophic ring of integers Z(I).