4 problems

Factorial-residue conjecture. It is conjectured that

Let $f_n$ be the polynomial defined in the paper and let $\operatorname{Spl}(f_n)$ denote its splitting field over $\mathbb{Q}$. Let $n_1\ne n_2$ be positive integers. Splitting-fi…

For each odd prime $p$, let $V(0,p-1)$ be the number of distinct residue classes modulo $p$ represented by $2!,3!,\dots,(p-1)!$. Erdős's asymptotic conjecture. As $p$ tends to infi…

For each odd prime $p$, let $V(H,N)$ denote the number of distinct residue classes modulo $p$ taken by the sequence $n!$ for $2\le n\le p-1$ and $H\le n\le H+N$. Erdős's conjecture…