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ON THE INVERSE IMAGES OF THE EULER FUNCTION $\varphi$
Keywords:
the Euler function, inverse image.DOI:
https://doi.org/10.17654/0972555524021Abstract
Thevalue $\varphi(m)$ for any positive integer $m$ of the Euler function $\varphi$ is equal to the number of integers $1 \leq a \leq m$ such that $\operatorname{gcd}(a, m)=1$.
In this article, based on data about the inverse image $\varphi^{-1}(n)$ for any even $2 \leq n \leq 5000$ calculated by the second author, we propose a conjecture asserting that for any even $n$, if the set $\varphi^{-1}(n)_{\text {o }}$ of odd inverse images of $n$ is not empty, then the minimum of $\varphi^{-1}(n)$ should be odd. Then we obtain some partial positive results on this conjecture. Moreover, we give some sufficient conditions for $n$ such that $\varphi^{-1}(n)$ or $\varphi^{-1}(n)_0$ is empty.
Received: February 13, 2024
Accepted: April 18, 2024
References
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M. W. Sierpinski, Voir solution du problem 230, Elem. Math. 11 (1956), 37.
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