Related papers: Long $n$-zero-free sequences in finite cyclic grou…

This paper describes a simple method for estimating lower bounds on the number of classes of equivalence for a special kind of integer sequences, called division sequences. The method is based on adding group structure to classes of…

In this paper, the $H\mathbb Z$-length of different groups is studied. By definition, this is the length of $H\mathbb Z$-localization tower or the length of transfinite lower central series of $H\mathbb Z$-localization. It is proved that,…

In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which…

We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull…

V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…

We study the problem of finding zero-sum blocks in bounded-sum sequences, which was introduced by Caro, Hansberg, and Montejano. Caro et al. determine the minimum $\{-1,1\}$-sequence length for when there exist $k$ consecutive terms that…

In this paper, we point out that the method used in [Acta Arith. 128(2007) 245-279] can be modified slightly to obtain the following result. Let $\varepsilon \in (0,\frac 14)$ and $c>0$, and let $p$ be a sufficiently large prime depending…

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of…

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding…

For integers a and n>0, let a(n) denote the residue class {x\in Z: x=a (mod n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1} does…

A set A of integers is said to be sum-free if there are no solutions to the equation x + y = z with x,y and z all in A. Answering a question of Cameron and Erdos, we show that the number of sum-free subsets of {1,...,N} is O(2^(N/2)).

Given an alphabet $S$, we consider the size of the subsets of the full sequence space $S^{\rm {\bf Z}}$ determined by the additional restriction that $x_i\not=x_{i+f(n)},\ i\in {\rm {\bf Z}},\ n\in {\rm {\bf N}}.$ Here $f$ is a positive,…

Let $G$ be a finite group multiplicatively written. The small Davenport constant of $G$ is the maximum positive integer ${\sf d}(G)$ such that there exists a sequence $S$ of length ${\sf d}(G)$ for which every subsequence of $S$ is…

A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the…

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a…

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can…

We construct a countable infinite graph G that does not contain cycles of length four having the property that the sequence of graphs $G_n$ induced by the first $n$ vertices has minimum degree $\delta(G_n)> n^{\sqrt{2}-1+o(1)}$.

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially…