Erdős Problem #970

declared status 'open'. Formalized: no. Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that$$h(k) \ll k^2?$$ Current best: Iwaniec [Iw78] proved$$h(k) \ll (k\log k)^2.$$The best lower bound known is$$h(k) \gg \frac{(\log k)(\log\log\log k)}{(\log\log k)^2}k,$$due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18]. Prize: no. OEIS: A048669. Tags: number theory.

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