Problem Statement
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\[h(k) \ll k^2?\]
Categories:
Number Theory
Progress
That $h(k)\ll k^2$ is a conjecture of Jacobsthal. 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].This is a more general form of the function considered in [687].
Source: erdosproblems.com/970 | Last verified: January 19, 2026