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.

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...

In the article the formal characterization of square-free numbers is shown; in this manner the paper is the continuation of [19]. Essentially, we prepared some lemmas for convenient work with numbers (including the proof that the sequence of prime reciprocals diverges [1]) according to [18] which were absent in the Mizar Mathematical Library. Some of them were expressed in terms of clusters’ registrations, enabling automatization machinery available in the Mizar system. Our main result of the article...

Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of ${\mathbb{Z}}_b=\mathbb{Z}/b\mathbb{Z}$, then the set $H_{\Gamma }=x\in \mathbb{N}|x+b\mathbb{Z}\in \Gamma \cup 1$ is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If $H_{\Gamma }$ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ 1, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM...

If $a$ and $b$ are positive integers with $a\le b$ and $a^2\equiv a\mathrm{mod}b$, then the set$$M_{a,b}=\{x\in \mathbb{N}:x\equiv a\mathrm{mod}b\text{or}x=1\}$$is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^{\times }$ and any $x\in M\setminus M^{\times }$ we say that $t\in \mathbb{N}$ is a factorization length of $x$ if and only if there exist irreducible elements $y_1,...,y_t$ of $M$ and $x=y_1\cdots y_t$. Let $ℒ\left(x\right)=\{t_1,...,t_j\}$ be the set of all such lengths (where $t_i\<t_{i+1}$ whenever $i\<j$). The Delta-set of the element $x$ is defined as the set of gaps in $ℒ\left(x\right)$: $\Delta \left(x\right)=\{t_{i+1}-t_i:1\le i\<k\}$ and the Delta-set of the monoid $M$ is given by ${\bigcup }_{x\in M\setminus M^{\times }}\Delta \left(x\right)$. We consider the $\Delta \left(M\right)$ when $M=M_{a,b}$ is an ACM with...

In this paper we investigate the solutions of the equation in the title, where $\phi$ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.

For an integer $n\ge 2$ we denote by $P\left(n\right)$ the largest prime factor of $n$. We obtain several upper bounds on the number of solutions of congruences of the form $n!+2^n-1\equiv 0(modq)$ and use these bounds to show that$$\underset{n\to \infty }{lim\; sup}P(n!+2^n-1)/n\ge (2{\pi }^2+3)/18.$$