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The Investigation of Euler’s Totient Function Preimages for φ(n)=2^mp1^αp2^β and the Cardinality of Pre-totients in General Case
Author(s) -
Ruslan Skuratovskii
Publication year - 2022
Publication title -
wseas transactions on mathematics
Language(s) - English
Resource type - Journals
eISSN - 2224-2880
pISSN - 1109-2769
DOI - 10.37394/23206.2022.21.7
Subject(s) - euler's totient function , mathematics , multiplicity (mathematics) , combinatorics , euler's formula , cardinality (data modeling) , sierpinski triangle , upper and lower bounds , function (biology) , discrete mathematics , mathematical analysis , fractal , evolutionary biology , biology , computer science , data mining
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.

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