Related papers: The structure of maximal zero-sum free Sequences

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption…

The $3k-4$ conjecture in groups $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}/p\mathbb{Z}$ satisfying $2A\neq \mathbb{Z}/p\mathbb{Z}$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-4\}$, then $A$ is…

Let $G$ be a finite abelian group and $S$ a sequence with elements of $G$. Let $|S|$ denote the length of $S$ and $\mathrm{supp}(S)$ the set of all the distinct terms in $S$. For an integer $k$ with $k\in [1, |S|]$, let $\Sigma_{k}(S)…

Kaplanski's Zero Divisor Conjecture envisions that for a torsion-free group G and an integral domain R, the group ring R[G] does not contain non-trivial zero divisors. We define the length of an element a in R[G] as the minimal non-negative…

A More Sums Than Differences (MSTD) set is a set of integers A contained in {0, ..., n-1} whose sumset A+A is larger than its difference set A-A. While it is known that as n tends to infinity a positive percentage of subsets of {0, ...,…

Let $G$ be a finite additive abelian group with exponent $d^kn, d,n>1,$ and $k$ a positive integer. For $S$ a sequence over $G$ and $A=\{1,2,\ldots,d^kn-1\}\setminus\{d^kn/d^i:i\in[1,k]\}, $ we investigate the lower bound of the number…

We investigate finite effect algebras and their classification. We show that an effect algebra with $n$ elements has at least $n-2$ and at most $(n-1)(n-2)/2$ nontrivial defined sums. We characterize finite effect algebras with these…

A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two elements in a poset are indistinguishable if they have the same strict up-set and the…

Let $G\cong C_{n_1}\oplus ... \oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1... w_n$ is a sequence of integers, all but at most one relatively prime to $|G|$,…

Let $m$ be a positive integer larger than $1$, let $w$ be a finite word over $\left\{0,1,...,m-1\right\}$ and let $a_{m;w}(n)$ be the number of occurrences of the word $w$ in the $m$-expansion of $n$ mod $p$ for any non-negative integer…

For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small,…

Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <=…

Let $\Gamma$ be a finite simple graph. If for some integer $n\geqslant 4$, $\Gamma$ is a $K_n$-free graph whose complement has an odd cycle of length at least $2n-5$, then we say that $\Gamma$ is an $n$-exact graph. For a finite group $G$,…

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…

Consider a subset [1,2,...,n]x[1,2,...,n] of the plane integer lattice. Take any non self-intersecting n^2-gon built on it (straight angles are allowed). The square of a side length is a positive integer. It is thus natural to ask how large…

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom…

This letter presents a direct construction of cross Z-complementary sequence sets (CZCSSs), whose aperiodic correlation sums exhibit zero correlation zones at both the front-end and tail-end shifts. CZCSS can be regarded as an extension of…

Let $G$ be a finite abelian group with exponent $n$. Let $\eta(G)$ denote the smallest integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a zero-sum subsequence of length at most $n$. We determine the precise…

A set is said to be \emph{3-free} if no three elements form an arithmetic progression. Given a 3-free set $A$ of integers $0=a_0<a_1<\cdots<a_t$, the \emph{Stanley sequence} $S(A)=\{a_n\}$ is defined using the greedy algorithm: For each…

The results of Bergelson-Host-Kra and Leibman say that a multiple polynomial correlation sequence can be decomposed into a sum of a nilsequence (a sequence defined by evaluating a continuous function along an orbit in a nilsystem) and a…