The value  for any positive integer m of the Euler function  is equal to the number of integers  such that  In this article, based on data about the inverse image  for any even  calculated by the second author, we propose a conjecture asserting that for any even n, if the set  of odd inverse images of n is not empty, then the minimum of  should be odd. Then we obtain some partial positive results on this conjecture. Moreover, we give some sufficient conditions for n such that  or  is empty.

the Euler function, inverse image.

Received: February 13, 2024; Accepted: April 18, 2024; Published: May 23, 2024

How to cite this article: Atsushi Yamagami and Yudai Sumiya, On the inverse images of the Euler function  JP Journal of Algebra, Number Theory and Applications 63(4) (2024), 335-362. https://doi.org/10.17654/0972555524021

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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