In the first part, we assign to each positive integer $n$ a digraph $\Gamma (n,5),$ whose set of vertices consists of elements of the ring ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ with the addition and the multiplication operations modulo $n,$ and for which there is a directed edge from $a$ to $b$ if and only if $a^5\equiv b(modn)$. Associated with $\Gamma (n,5)$ are two disjoint subdigraphs: ${\Gamma }_1(n,5)$ and ${\Gamma }_2(n,5)$ whose union is $\Gamma (n,5).$ The vertices of ${\Gamma }_1(n,5)$ are coprime to $n,$ and the vertices of ${\Gamma }_2(n,5)$ are not coprime to $n.$ In this part, we study the structure of $\Gamma (n,5)$ in detail. In the second part, we investigate...

A sequence is called $k$-automatic if the $n$’th term in the sequence can be generated by a finite state machine, reading $n$ in base $k$ as input. We show that for many multiplicative functions, the sequence ${\left(f\left(n\right)\text{mod}v\right)}_{n\ge 1}$ is not $k$-automatic. Among these multiplicative functions are ${\gamma }_m\left(n\right),{\sigma }_m\left(n\right),\mu \left(n\right)$ et $\phi \left(n\right)$.

We examine iteration graphs of the squaring function on the rings $\mathbb{Z}/n\mathbb{Z}$ when $n=2^{k}p$, for $p$ a Fermat prime. We describe several invariants associated to these graphs and use them to prove that the graphs are not symmetric when $k=3$ and when $k\ge 5$ and are symmetric when $k=4$.

The paper extends the results given by M. Křížek and L. Somer, , Czech. Math. J. 54 (129) (2004), 465–485 (see [5]). For each positive integer $n$ define a digraph $\Gamma \left(n\right)$ whose set of vertices is the set $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^3\equiv b(modn).$ The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph $\Gamma \left(n\right)$ is proved. The formula for the number of fixed points of $\Gamma \left(n\right)$ is established. Moreover, some connection of the length...

For $1\le c\le p-1$, let $E_1,E_2,\cdots ,E_m$ be fixed numbers of the set $\{0,1\}$, and let $a_1,a_2,\cdots ,a_m$ $(1\le a_i\le p$, $i=1,2,\cdots ,m)$ be of opposite parity with $E_1,E_2,\cdots ,E_m$ respectively such that $a_1{a}_2\cdots a_m\equiv c(modp)$. Let $$N(c,m,p)=\frac{1}{2^{m-1}}\underset{a_1{a}_2\cdots a_m\equiv c(modp)}{\sum _{a_1=1}^{p-1}\sum _{a_2=1}^{p-1}\cdots \sum _{a_m=1}^{p-1}}(1-{(-1)}^{a_1+E_1})(1-{(-1)}^{a_2+E_2})\cdots (1-{(-1)}^{a_m+E_m}).$$ We are interested in the mean value of the sums $$\sum _{c=1}^{p-1}{E}^2(c,m,p),$$ where $E(c,m,p)=N(c,m,p)-\left({(p-1)}^{m-1}\right)/\left(2^{m-1}\right)$ for the odd prime $p$ and any integers $m\ge 2$. When $m=2$, $c=1$, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.