Related papers: Erd\H{o}s covering systems
Introduced by Erd\H{o}s in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over…
A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erd\H{o}s in 1950, who asked whether…
The minimum modulus problem on covering systems was posed by Erd\H{o}s in 1950, who asked whether the minimum modulus of a covering system with distinct moduli is bounded. In 2007, Filaseta, Ford, Konyagin, Pomerance and Yu affirmed it if…
In 1950, Erd\H{o}s posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was…
Since their introduction by Erd\H{o}s in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding…
A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erd\H{o}s in 1950, and over the following decades numerous problems…
Covering systems were introduced by Erd\H{o}s in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long…
Erd\H{o}s first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in…
A covering system of the integers is a finite collection of modular residue classes $\{a_m \bmod{m}\}_{m \in S}$ whose union is all integers. Given a finite set $S$ of moduli, it is often difficult to tell whether there is a choice of…
Covering systems of the integers were introduced by Erd\H{o}s in 1950. Since then, many beautiful questions and conjectures about these objects have been posed. Most famously, Erd\H{o}s asked whether the minimum modulus of a covering system…
The concept of a covering system was first introduced by Erd\H{o}s in 1950. Since their introduction, a lot of the research regarding covering systems has focused on the existence of covering systems with certain restrictions on the moduli.…
A $\textit{covering system}$ is a collection of integer congruences such that every integer satisfies at least one congruence in the collection. A covering system is called $\textit{distinct}$ if all of its moduli are distinct. An expansive…
We answer a question of Erd\H{o}s by showing that the least modulus of a distinct covering system of congruences is no larger than $10^{18}$.
In 1952, H. Davenport posed the problem of determining a condition on the minimum modulus $m_{0}$ in a finite distinct covering system that would imply that the sum of the reciprocals of the moduli in the covering system is bounded away…
A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most…
Erd\H{o}s and Graham (Erd\H{o}s and Graham, 1980) asked if there exists an $n$ such that the divisors of $n$ greater than 1 are the moduli of a distinct covering system with the following property: If there exists an integer which satisfies…
A set of arithmetical sequences $$ a_1\, (\bmod{ \,\, m_1}) \quad, \quad a_2 \, (\bmod{\,\, m_2}) \quad, \quad \dots \quad , \quad a_k \, (\bmod{\,\,m_k}) \quad \quad , $$ with $$ m_1 \leq m_2 \leq \dots \leq m_k \quad \quad , $$ is called…
Bob Hough recently disproved a long-standing conjecture of Paul Erd\H{o}s regarding covering systems. Inspired by his seminal paper, we describe analogs of covering systems to Boolean functions, and more generally, the problem of covering…
The covering radius problem is a question in coding theory concerned with finding the minimum radius $r$ such that, given a code that is a subset of an underlying metric space, balls of radius $r$ over its code words cover the entire metric…
We prove that every distinct covering system has a modulus divisible by either 2 or 3.