We discuss a group-theoretical generalization of the well-known Gauss formula involving the function that counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.

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...$

The number of solutions of the congruence $a₁x₁+\cdots +a_k{x}_k\equiv 0\left(modn\right)$ in the box $0\le x_i\le b_i$ is estimated from below in the best possible way, provided for all i,j either $(a_i,n)|(a_j,n)$ or $(a_j,n)|(a_i,n)$ or $n\left|\right[a_i,a_j]$.

We answer a question of Bednarek proposed at the 9th Polish, Slovak and Czech conference in Number Theory.

The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

Let $\mathbb{N}$ be the set of positive integers and let $s\in \mathbb{N}$. We denote by $d^s$ the arithmetic function given by $d^s\left(n\right)={\left(d\left(n\right)\right)}^s$, where $d\left(n\right)$ is the number of positive divisors of $n$. Moreover, for every $\ell ,m\in \mathbb{N}$ we denote by ${\delta }^{s,\ell ,m}\left(n\right)$ the sequence $$\underset{m\text{-times}}{\underbrace{d^s(d^s(...d^s(d^s\left(n\right)+\ell )+\ell ...)+\ell )}}=\left\{\begin{array}{cc}{d}^s\left(n\right)\hfill & \text{for}m=1,\hfill \\ d^s(d^s\left(n\right)+\ell )\hfill & \text{for}m=2,\hfill \\ d^s(d^s(d^s\left(n\right)+\ell )+\ell )\hfill & \text{for}m=3,\hfill \\ ⋮\hfill \end{array}\right.$$ We present classical and nonclassical notes on the sequence ${\left({\delta }^{s,\ell ,m}\left(n\right)\right)}_{m\ge 1}$, where $\ell$, $n$, $s$ are understood as parameters.

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions.We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form ${\prod }_{p\in B}p-{\epsilon }_0$ for...