30 problems

Put … where the maximum is taken over all disjoint systems for which $n_1 > m$. Determine or estimate $\varepsilon_m$ as well as possible. I could not even decide whether…

Suppose $n_1 < n_2 < \dots$ are such that for every choice of $a_i$ the set of integers not satisfying any of the congruences $a_i \pmod{n_i}$ has density 0. In this case we must h…

Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one…

Let $r\in\mathbb N$, let $a_1,\dots,a_r\in\mathbb Z$, and let $n_1,\dots,n_r\in\mathbb N$, with the moduli not required to be distinct. If there exists an integer $k$ such that eve…

Does there exist a natural number $n$ and a function $a:\mathbb{N}\to\mathbb{Z}$ such that, writing $D=\{d\in\mathbb{N}:d\mid n\text{ and }d>1\}$, both of the following hold? Every…

Call distinct integers $1<n_1<\cdots<n_k$ an irreducible covering set if suitable residue classes modulo them cover every integer, but no proper subset does. Determine the number o…

A congruence class $(n,a)$ represents the integers congruent to $a\pmod n$, where $2\le n$ and $0\le a<n$. A finite set of such classes is a minimal distinct covering system if eve…

Many further unsolved problems can be asked about covering systems. Selfridge and I asked: Is there a covering system all whose moduli are odd?

Let $n,a:\mathbb{N}\to\mathbb{N}$ satisfy that $n$ is strictly increasing and $a(i)<n(i)$ whenever $i\geq1$. For $k\in\mathbb{N}$, let $U_k$ be the number of natural numbers…

A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a covering set for $m$ if every…

There is no finite family of congruence classes $a_i\pmod{n_i}$ with pairwise distinct moduli $n_i$ such that every integer belongs to exactly one of the classes.

Does there exist a covering system of the integers in which no modulus divides any other modulus?

Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and…

Let $A=\{n_1<\cdots<n_r\}$ be a finite set of positive integers. What is the maximum density of integers covered by a suitable choice of congruences $a_i\pmod{n_i}$? Is the minimum…

The following extremal problem can now be posed: Put … where the maximum is to be taken over all the $m$ for which the divisors of $m$ do not form a covering system. By Haight's th…

Is it possible for a Lucas sequence to have all terms composite $\textit{without}$ having an underlying system of covering congruences responsible? (In other words, no positive int…

If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)

Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?

Is there an integer $m\geq 1$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$?

Let us now restrict ourselves to a special case. The $n_i$ are the integers between $t$ and $ct$. First of all denote by $\alpha_1(c,t)$ the smallest possible value of the density…

A family of residue classes $a_i \pmod{n_i}$ with $1 < n_1 < \ldots < n_r$ is called a system of covering congruences if every integer belongs to at least one of the residue classe…

A system of congruences … is called a covering system if every integer satisfies at least one of the congruences in (3). The simplest covering system is $0 \pmod 2$, $0 \pmod 3$,…

Erdős–Selfridge odd covering problem. Does there exist a covering system of $\mathbb{Z}$ whose moduli are odd, distinct, and greater than $1$?

A distinct covering system is a covering system whose moduli are distinct integers greater than $1$. Let $m\geq 2$, let $a$, $b$, and $c$ be positive integers, and let $L$ denote t…

Lemma nice criterion. If $p$ is the smallest prime divisor of $n$ and $n/p$ has fewer than $p$ distinct prime divisors, then $n$ is non-intersecting.