The investigation of Euler's totient function preimages

Abstract

We propose a lower estimation for computing quantity of the inverses of Euler's function. We answer the question about the multiplicity of $m$ in the equation $\varphi(x) = m$ \cite{Ford}. An analytic expression for exact multiplicity of $m = {{2}^{{{2}^{n}}+a}}$, where $a\in N$, $a<{{2}^{n}}$, $\varphi(t)={{2}^{{{2}^{n}}+a}}$ was obtained. A lower bound of inverses number for arbitrary $m$ was found. New numerical metric was proposed.