The Investigation of Euler’s Totient Function Preimages for φ(n)=2^mp1^αp2^β and the Cardinality of Pre-totients in General Case

Abstract

This paper shows how to determine all those positive integers x such that φ(x) = m holds, where x is of the form 2^αp^bq^c and p, q are distinct odd primes and a, b, c ∈ N. In this paper, we have shown how to determine all those positive integers n such that φ(x) = n will hold where n is of the form 2^αp^bq^c where p, q are distinct odd primes and a, b, c ∈ N. Such n are called pre-totient values of 2^αp^bq^c. Several important theorems along with subsequent results have been demonstrated through illustrative examples. We propose a lower bound for computing quantity of the inverses of Euler’s function. We answer the question about the multiplicity of m in the equation φ(x) = m [1]. An analytic expression for exact multiplicity of m= 2^2n+a where a ∈ N, α < 2^n, φ(x) = 2^2n+a was obtained. A lower bound of inverses number for arbitrary m was found. We make an approach to Sierpinski assertion from new side. New numerical metric was proposed.