Jacobsthal's quadratic-bound question: is h(k) = O(k^2)?

Theorem 1.1: $h(k)\le Ck^2/(\log\log(3k))^2$ for an absolute constant $C$ and every $k\ge1$, uniformly over prime sets and interval positions; in particular $h(k)\ll k^2$, answering Jacobsthal's question and improving Iwaniec's $k^2\log^2k$. The proof pushes the linear sieve to its limiting parameter, keeping a boundary term, and controls the discrepancy with the actual residue classes through an inverse estimate and stopped counts. The constant is not explicit, and the true order of $h(k)$ (Erdos #970's broader ask) is not determined: the gap to the lower bound near $k(\log k)^2$ remains. Theorem 1.2 of the same paper also bounds the quantity in Erdős Problem #687: every choice of one class per prime $p\le z$ leaves survivors below $z^2/(\log z)^2$, so $Y(z)<z^2/(\log z)^2$; the connection was pointed out in a VibeMathed submission on 14 October 2026.

Theorem 1.1: $h(k)\le Ck^2/(\log\log(3k))^2$ for an absolute constant $C$ and every $k\ge1$, uniformly over prime sets and interval positions; in particular $h(k)\ll k^2$, answering Jacobsthal's question and improving Iwaniec's $k^2\log^2k$. The proof pushes the linear sieve to its limiting parameter, keeping a boundary term, and controls the discrepancy with the actual residue classes through an inverse estimate and stopped counts. The constant is not explicit, and the true order of $h(k)$ (Erdos #970's broader ask) is not determined: the gap to the lower bound near $k(\log k)^2$ remains. Theorem 1.2 of the same paper also bounds the quantity in Erdős Problem #687: every choice of one class per prime $p\le z$ leaves survivors below $z^2/(\log z)^2$, so $Y(z)<z^2/(\log z)^2$; the connection was pointed out in a VibeMathed submission on 14 October 2026.