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Gholam Reza Pourgholi

Publications and source records attributed to Gholam Reza Pourgholi.

4 recordsLinked to original sources
arXiv2015

On The Lehmer Numbers, I

A composite number $n$ is called Lehmer when $ϕ(n) | n - 1$, where $ϕ$ is the Euler totient function. In 1932, D.~H.~Lehmer conjectured that there are no composite Lehmer numbers and showed that Lehmer numbers must be odd and square-free. Although a number of additional constraints have been found since, the problem remains still open. For each odd number $m>1$, let $m^\star$ be the largest number such that $2^{m^\star}$ divides $m-1$. Using this notion we present some new necessary conditions and introduce a method to construct some new family of numbers $n$ which are not Lehmer number.

math.NT↗
arXiv2015

On the Lehmer Numbers, II

A composite number $n$ is called a Lehmer number when $ϕ(n) | n - 1$, where $ϕ$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $ϕ(n)| n-1$? No such numbers are known. In this paper we establish an Euler Totient Inequality and relate some new parameters to Lehmer numbers. As an application of what we have done, we show that for all prime numbers $p$ and for all odd square-free numbers $n$, at most one of $n$ or $pn$ is a Lehmer number. Finally we suggest some open problems for the future investigations on the Lehmer numbers.

math.NT↗
arXiv2014

A group sum inequality and its application to power graphs

Let $G$ be a finite group of order $n$, and let $C_n$ be the cyclic group of order $n$. We show that $\sum_{g \in C_n} ϕ(\mathrm{o}(g))\geq \sum_{g \in G} ϕ(\mathrm{o}(g))$, with equality if and only if $G$ is isomorphic to $C_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of undirected edges in its directed power graph.

math.GR↗