Some remarks on Euler's totient function

TL;DR

This paper investigates the structure of the preimages of even numbers under Euler's totient function, analyzing which even numbers are in its image and their preimage properties.

Contribution

It provides new insights into the structure of preimages of certain numbers in the image of Euler's totient function.

Findings

Not all even numbers are in the image of the totient function.

Preimages of in-image even numbers can contain only even numbers.

The preimage structure varies depending on the specific even number considered.

Abstract

The image of Euler's totient function is composed of the number 1 and even numbers. However, many even numbers are not in the image. We consider the problem of finding those even numbers which are in the image and those which are not. If an even number is in the image, then its preimage can have at most half its elements odd. However, it may contain only even numbers. We consider the structure of the preimage of certain numbers in the image of the totient function.