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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.

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A quadratic bound for Jacobsthal's function

Sep 25, 2026·paper.pdfLean

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.

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