Related papers: Divisibility among power GCD and power LCM matrice…

Fix a partition $\mu=(\mu_1,\dotsc,\mu_m)$ of an integer $k$ and positive integer $d$. For each $n>k$, let $\chi^\lambda_\mu$ denote the value of the irreducible character of $S_n$ at a permutation with cycle type…

We define $g(n)$ to be the maximal order of an element of the symmetric group on $n$ elements. Results about the prime factorization of $g(n)$ allow a reduction of the upper bound on the largest prime divisor of $g(n)$ to $1.328\sqrt{n\log…

A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. This matrix is always unimodular so that the inverse g of L has integer entries g(x,y). In…

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n -…

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $\rho_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic…

The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b. Let P(n) be the largest prime factor of n, S(x,y) = {n<=x: P(n) <= y} and Psi(x,y) = Card S(x,y). Let…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

We construct a new arithmetic-term representation for the function gcd(a,b). As a byproduct, we also deduce a representation gcd(a,b) by a modular term in integer arithmetic.

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that…

A notion of gcd chain has been introduced by the author at ISSAC 2017 for two univariate monic polynomials with coefficients in a ring R = k[x_1, ..., x_n ]/(T) where T is a primary triangular set of dimension zero. A complete algorithm to…

We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. The splitting is called {\it…

Let $G$ be a linear algebraic group defined over an algebraically closed field $k$, and let $V$ be a vector space on which $G$ acts linearly. The separating variety $\mathcal{S}_{G,V}$ is the subvariety of $V^2$ consisting of pairs of…

Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $\Delta$-bounded if the diameter of each cluster is bounded by $\Delta$. A distribution over $\Delta$-bounded partitions is a $\beta$-padded decomposition if every ball of radius…

A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup…

We describe how to compute for two polynomials $f(X), g(X) \in \mathbb{Z}[X]$ with integer coefficients the greatest common divisors of $f(z)$ and $g(z)$ for all integers $z \in \mathbb{Z}$. As an application we determine the structures…

For each integer k >= 2, we determine a sharp bound on mad(G) such that V(G) can be partitioned into sets I and F_k, where I is an independent set and G[F_k] is a forest in which each component has at most k vertices. For each k we…

If $G$ is an abelian group, we say $S\subset G$ is a set of recurrence if for every probability measure preserving $G$-system $(X,\mu,T)$ and every $D\subset X$ having $\mu(D)>0$, there is a $g\in S$ such that $\mu(D\cap T^{g}D)>0$. We say…

Let m_1,...,m_s be positive integers. Consider the sequence defined by multinomial coefficients: a_n=\binom{(m_1+m_2+... +m_s)n}{m_1 n, m_2 n,..., m_s n}. Fix a positive integer k\ge 2. We show that there exists a positive integer C(k) such…

For a group G and an element a in G let |a|_k denote the cardinality of the set of commutators [a,x_1,...,x_k], where x_1,...,x_k range over G. The main result of the paper states that a group G is finite-by-nilpotent if and only if there…