| # A quadratic bound for Jacobsthal’s function | |
| The following describes the scope of the Lean formalization related to the following accompanying paper(s): | |
| - [A quadratic bound for Jacobsthal's function](../../preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf) | |
| ## Scope | |
| Let $h(k)$ be the least interval length that guarantees an integer coprime to any prescribed positive modulus with at most $k$ distinct prime factors. The formalization proves the paper's strengthened Jacobsthal bound $h(k)\le Ck^2/(\log\log(3k))^2$ for one absolute $C>0$ and every $k\ge1$. Intervals may begin at any signed integer. The earlier quadratic bound is also retained as a separate statement; the optimal order of growth is not determined. | |
| ## Comparator links | |
| | Result | Comparator statement | | |
| | --- | --- | | |
| | Quadratic Jacobsthal bound | [Jacobsthal.lean](../ComparatorChallenges/Jacobsthal.lean) | | |
| | Jacobsthal bound with an iterated-logarithm saving | [JacobsthalImproved.lean](../ComparatorChallenges/JacobsthalImproved.lean) | | |