8 problems

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…

Erdős's stronger conjecture. There exist integers $k$ and $0<x<y$ with $y>x+k$ such that

Sah's conjecture. As $N\to\infty$,

Let $q=p^k$ be a prime power, and let $f\in\mathbb{F}_q[T][X]$ be a fixed irreducible polynomial with $X$-degree $d\geq 2$. Define … Let $V_f:=\{g\in\mathbb{F}_q[T]:f(X+g)=f(X)\}$…

The conjecture for $\psi_F$. One should have

Let $f\in\mathbb Z[x]$ be irreducible over $\mathbb Q$ of degree $d\ge 2$, and define … where $\operatorname{rad}(n)$ is the product of the distinct primes dividing $n$. Radical LC…

Let $f\in\mathbb Z[x]$ be irreducible over $\mathbb Q$ of degree $d\ge 2$, and define … ignoring zero values and taking the least common multiple of an empty set to be $1$. Cilleru…

Cilleruelo's conjecture. As $N\to\infty$,