Euler Phi Function for Modular Factorizations

Number theory is a branch of mathematics that studies the integers. There are people well known characters that studied the numbers such as Pythagoras, Euler, Fermat, and others. For the eighteenth century, one of the greatest mathematicians, Leonhard Euler, contributes to the definition of the Euler Phi Function, which is an example of a multiplicative arithmetic function.

In number theory, arithmetic functions are always considered to create some interesting properties. Including the Euler Phi Function, the function that counts the number smaller than, and relatively prime factors the number in question. This project aims to define the same idea but in the theory of τ_n-factorization or modular factorizations. It is known that the τ₁-factorization are the usual factorizations, therefore the usual Euler Phi Function coincides with the one defined for the τ₁-factorization theory.

There have been more closely at some patterns for τ_n-factorizations for n=0 and n=2. This is a work in progress, which attempts to extend and understand this notion for n≥3.