OpenAI · 021-01 · 2026-09-25 · Number theory
A quadratic bound for Jacobsthal's function
preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026
Family 021. A quadratic bound for Jacobsthal's function.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Answers Jacobsthal's quadratic-bound question: every interval of consecutive integers contains an integer coprime to any prescribed positive integer with at most distinct prime divisors, for an absolute constant . The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.
Let be the least integer such that every interval of consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove , giving an affirmative answer to Jacobsthal's quadratic-bound question.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 021: A quadratic bound for Jacobsthal’s function. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.