OPEN
Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and
\[p(n+k)>k^2,\]
where $p(m)$ is the least prime factor of $m$?
Related to questions of Erdős, Eggleton, and Selfridge. This may be true with $k^2$ replaced by $k^d$ for any $d$.
See also [680] and [682].
