Related papers: A Multiplicative Property for Zero-Sums I

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd}…

Let $G$ be a finite abelian group of order $n$ and let $\Delta_{n-1}$ denote the $(n-1)$-simplex on the vertex set $G$. The sum complex $X_{A,k}$ associated to a subset $A \subset G$ and $k < n$, is the $k$-dimensional simplicial complex…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

Let $\mathbb{F}$ be an infinite field with characteristic different from two. For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;\mathbb{F})$ be the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ over $\mathbb{F}$ with…

The \textit{sepr-sequence} of an $n\times n$ real matrix $A$ is $(s_1,\ldots,s_n)$, where $s_k$ is the subset of those signs of $+,-,0$ that appear in the values of the $k\times k$ principal minors of $A$. The $12\times 12$ matrix…

Let $G$ be an additive finite abelian group, and let $\mathrm{disc}(G)$ denote the smallest positive integer $t$ with the property that every sequence $S$ over $G$ with length $|S|\geq t $ contains two nonempty zero-sum subsequences of…

For a finite group $G$, we denote by ${\sf d}(G)$ and by ${\sf E}(G)$, respectively, the small Davenport constant and the Gao constant of $G$. Let $C_n$ be the cyclic group of order $n$ and let $G_{m,n,s} = C_n \rtimes_s C_m$ be a…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful.…

Let $D(G)$ be the Davenport constant of a finite Abelian group $G$. For a positive integer $m$ (the case $m = 1$, is the classical one) let ${\mathsf E}_m(G)$ (or $\eta_m(G)$, respectively) be the least positive integer $t$ such that every…

Let $A,B\subseteq\mathbb Z_n$ be given and $S=(x_1,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that…

An arithmetical structure on the complete graph $K_n$ with $n$ vertices is given by a collection of $n$ positive integers with no common factor each of which divides their sum. We show that, for all positive integers $c$ less than a certain…

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…

The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the…

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$ with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the…

Given a finite commutative semigroup $\mathcal{S}$ (written additively), denoted by ${\rm D}(\mathcal{S})$ the Davenport constant of $\mathcal{S}$, namely the least positive integer $\ell$ such that for any $\ell$ elements…

In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\mathcal{H}(x) = \{gx + h_j\}_{j=1}^k$ of linear forms in…

Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

Let $\mathbb Z_n$ be the cyclic group of order $n \ge 3$ additively written. S. Savchev \& F. Chen (2007) proved that for each zero-sum free sequence $S = a_1 \bullet \dots \bullet a_t$ over $\mathbb Z_n$ of length $t > n/2$, there is an…