47 problems

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…

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

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 $1$?

Bogatyi–Shavgulidze's conjecture. Every covering system has a periodic point of period at most $k$ in $I_1\cup\cdots\cup I_k$. This conjecture is the analogue, for covering systems…

A positive integer $n$ is a covering number if there are distinct divisors $n_1,\ldots,n_k>1$ of $n$ and integers $a_1,\ldots,a_k$ such that $\mathbb Z$ is the union of the residue…

Fix a real number $K>1$. A covering system is a finite set of congruences covering every integer, and its moduli are distinct. Erdős–Graham bounded-interval conjecture. For suffici…

Fix a real number $K>1$. For sufficiently large $N$, choose arbitrary integers $r(n)$ for each modulus $n\in(N,KN]$, using the corresponding residue classes $r(n)\pmod n$. Erdős–Gr…

A covering system is a finite set of congruences such that every integer satisfies at least one congruence; here the moduli are required to be distinct. For a covering system, writ…

Power integral basis removal conjecture. The requirement that $K$ have a power integral basis can be cancelled: the displayed set is empty or has at least $2^m$ elements.

Znam–Korec conjecture. One has

A system of residue classes is a finite collection $A=\{a_s(n_s)\}_{s=1}^k$ of residue classes of $\mathbb Z$, where $a_s(n_s)=a_s+n_s\mathbb Z$ and the moduli $n_1,\ldots,n_k$ are…

Let $q,m\geq 3$. Let $p_1,p_2,\ldots,p_m$ be distinct integers satisfying $0<p_k<q$, $\gcd(q,p_1,\ldots,p_m)=1$, and such that no proper subset $I\subsetneq\{1,\ldots,m\}$ satisfie…

For a finite system $A=\{a_s(n_s)\}_{s=1}^k$ of arithmetic sequences, where $a_s(n_s)=a_s+n_s\mathbb Z$, suppose that the system is disjoint, meaning that its sequences are pairwis…

Burshtein's conjecture. One has

Let … be a finite cover of the integers by residue classes. A cover is a partition if these residue classes are pairwise disjoint. Erdős's conjecture. The system $A$ cannot be a pa…

For a positive integer $m$, let $L_{\min}(m)$ be the smallest possible least common multiple among all distinct covering systems whose minimum modulus is $m$. Klein's conjecture. T…

A distinct covering system is a covering system whose moduli are distinct integers greater than $1$. Let $m$ be its least modulus and let $L$ be the least common multiple of its mo…

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…

A distinct covering system is a covering system whose moduli are distinct integers greater than $1$. Let $m$ be its least modulus and $M$ its largest modulus. Klein's conjecture. I…

A distinct covering system is a covering system whose moduli are distinct integers greater than $1$. Let $m$ be its least modulus and $M$ its largest modulus. Krukenberg's conjectu…

Fix a multiplicity $m$, and let a disjoint covering system (DCS) satisfy the normalization condition … Let … be its largest modulus. Two-prime extremal modulus conjecture. Among DC…

Let a disjoint covering system (DCS) have multiplicity $m$, and impose the normalization condition … with precisely one repeated modulus. Finiteness conjecture. For each fixed $m$,…

Minimum-modulus 7 least-common-multiple conjecture. If the smallest modulus of a covering system with distinct moduli is $7$, then