12 problems

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.

Let $R(s,q)$ denote the minimum size of a covering system of vectors over $Z_s^q$. In particular, consider the case $s=3$ as $q$ tends to infinity. The $s=3$ asymptotic conjecture.…

Let $R(s,q)$ denote the minimum size of a covering system of vectors over $Z_s^q$. For fixed $s\geq 3$, consider the growth of $R(s,q)$ as $q$ tends to infinity. Asymptotic growth…

Fraenkel's conjecture. If the system has no multiplicity, then $\alpha_i=p/q_i$, where

Dynamically defined covering-systems conjecture. These choices can satisfy both $\max_i\|n_my_i-w_i\|\leq\varepsilon/2$ for every $m=1,\dots,M$, and

Let $L$ be a primitive covering number, meaning a positive integer that is a covering number but has no proper divisor that is a covering number. Write its prime factorization as ……

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…

Let a covering system be a finite collection of residue classes $C=\{a_m\bmod{m}\}_{m\in S}$ with distinct moduli $S\subset\mathbb{N}_{>1}$ whose union is $\mathbb{Z}$. Erdős's Odd…

Let $a_i \pmod{n_i}$, with $n_1<n_2<\cdots<n_k$, be a finite system of congruences. Call it a covering system if every integer satisfies at least one of the congruences. Is it true…