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OPEN This is open, and cannot be resolved with a finite computation.
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 $m$?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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].

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Formalised statement? Yes
Related OEIS sequences: A389680
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #681, https://www.erdosproblems.com/681, accessed 2026-01-16