The investigation of Euler's totient function preimages

TL;DR

This paper provides new lower bounds and exact formulas for the number of solutions to Euler's totient function equation, enhancing understanding of its inverse images and their multiplicities.

Contribution

It introduces a lower estimation method for counting inverses of Euler's totient function and derives an exact formula for specific exponential forms of m.

Findings

Derived an exact multiplicity formula for m = 2^{2^n + a}

Established a lower bound for the number of inverses for arbitrary m

Proposed a new numerical metric for analyzing totient 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.