A quadratic bound for Jacobsthal’s function

Answers Jacobsthal's quadratic-bound question: every interval of Ck^2 consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss. (Lean)

Manuscript

A quadratic bound for Jacobsthal's function

September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026}

Let h(k) be the least integer such that every interval of h(k) consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove h(k)\ll k^2/(\log\log(3k))^2, giving an affirmative answer to Jacobsthal's quadratic-bound question.

Result 021 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.