Some remarks on Euler's totient function

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.