WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 21, 2022
The Investigation of Euler’s Totient Function Preimages for $$φ(n)=2^{m}p1^{α}p2^{β}$$ and the Cardinality of Pre-totients in General Case
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Abstract: This paper shows how to determine all those positive integers x such that $$φ(x) = m$$ holds, where $$x$$ is of the form $$2^{α}p^{b}q^{c}$$ and $$p, q$$ are distinct odd primes and $$a, b, c \in\mathbb{N}$$. 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 $$2^{α}p^{b}q^{c}$$ where $$p, q$$ are distinct odd primes and $$a, b, c \in\mathbb{N}$$. Such $$n$$ are called pre-totient values of $$2^{α}p^{b}q^{c}$$. 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 $$\in\ N, α < 2^n, φ(x) = 2^{2n+a}$$ 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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Pages: 44-52
DOI: 10.37394/23206.2022.21.7