Related papers: On the minimal Hamming weight of a multi-base repr…

Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a…

We consider the complexity for computing the approximate sum $a_1+a_2+...+a_n$ of a sorted list of numbers $a_1\le a_2\le ...\le a_n$. We show an algorithm that computes an $(1+\epsilon)$-approximation for the sum of a sorted list of…

A vertex subset M of a graph G is a multipacking if for each vertex v, and each positive integer s less than or equal to the diameter of G, v is within distance s of at most s vertices of M. The multipacking number of a graph is the maximum…

A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…

Notice that the square of $9376$ is $87909376$ which has as its rightmost four digits $9376$. To generalize this remarkable fact, we show that, for each integer $n\ge 2$, there exists at least one and at most two positive integers $x$ with…

In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor…

Our main interest is the low-rank approximation of a matrix in R^m.n under a weighted Frobenius norm. This norm associates a weight to each of the (m x n) matrix entries. We conjecture that the number of approximations is at most min(m, n).…

Let $\omega^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}\omega^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent…

Fix a positive integer $n$ and consider the bipartite graph whose vertices are the $3$-element subsets and the $2$-element subsets of $[n]=\{1,2,\dots,n\}$, and there is an edge between $A$ and $B$ if $A\subset B$. We prove that the…

We consider the problem of maximizing a non-negative monotone submodular function subject to a knapsack constraint, which is also known as the Budgeted Submodular Maximization (BSM) problem. Sviridenko (2004) showed that by guessing 3…

We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an…

Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the reversed…

A base-$g$ Niven number is a natural number divisible by the sum of its base-$g$ digits. We show that, for any $g\geq 3$, all sufficiently large natural numbers can be written as the sum of three base-$g$ Niven numbers. We also give an…

Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent…

As techniques for graph query processing mature, the need for optimization is increasingly becoming an imperative. Indices are one of the key ingredients toward efficient query processing strategies via cost-based optimization. Due to the…

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it take Omega(n^2) time,…

The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. Begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is…

In this article, we investigate the multi-parametric matroid problem. The weights of the elements of the matroid's ground set depend linearly on an arbitrary but fixed number of parameters, each of which is taken from a real interval. The…

The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first…

In a quantum system having a finite number $N$ of orthogonal states, two orthonormal bases $\{a_i\}$ and $\{b_j\}$ are called mutually unbiased if all inner products $<a_i|b_j>$ have the same modulus $1/\sqrt{N}$. This concept appears in…