For any positive integer n let ϕ(n) and σ(n) be the Euler function of n and the sum of divisors of n, respectively. In [5], Mąkowski and Schinzel conjectured that the inequality σ(ϕ(n)) ≥ n/2 holds for all positive integers n. We show that the lower density of the set of positive integers satisfying the above inequality is at least 0.74.

We assign to each positive integer $n$ a digraph whose set of vertices is $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\left(modn\right)$. We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

Let $p_1,p_2,\cdots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k\ge 5$, then $$(p_{k+1}-1)!\mid \left(\frac{1}{2}(p_{k+1}-1)\right)!p_k!,$$ which improves a previous result of the second author.

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.