A quadratic bound for Jacobsthal’s function

What OpenAI claims

Answers Jacobsthal's quadratic-bound question: every interval of Ck² 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)

Paper

1. A quadratic bound for Jacobsthal's function

   25 September 2026 · Preprint, not peer reviewed · PDF

   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) ≪ k²/(log log(3k))², giving an affirmative answer to Jacobsthal's quadratic-bound question.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.