In 2023, M. Drmota, C. M\"{u}llner and I showed that for $c$ in $(1,2)$ and $H$ in $\mathbb{N}$, the set $\mathcal{N}(c,H)$ consisting of the integers $n$ such that all the numbers

$\lfloor (n+1)^c\rfloor, \lfloor (n+2)^c\rfloor, \cdots, \lfloor (n+H)^c\rfloor \, \text{ are pairwise coprime,}$

is infinite. I'll present parts of a joint work in progress with H. Iwaniec, which implies that for any $c$ and $H$ as above, the set $\mathcal{N}(c,H)$ has a positive asymptotic density.