The board / number theory / #970
Erdős Problem #970
Jacobsthal's function
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`IsJacobsthalBound k m` says that every interval of `m` consecutive integers contains an integer coprime to every positive natural number having at most `k` distinct prime factors.
Jacobsthal's function, uniformly parametrized by the maximum number of distinct prime factors.
Let \(h(k)\) be Jacobsthal's function, defined to as the minimal \(m\) such that, if \(n\) has at most \(k\) prime factors, then in any set of \(m\) consecutive integers there exists an integer coprime to \(n\). Determine the order of magnitude of \(h(k)\). In particular, is it true that \[h(k) \ll k^2?\] The bound \(h(k) \ll k^2\) was proved by an internal OpenAI model in September 2026 [OAI26a]; see the linked Lean proof. The order of magnitude of \(h(k)\) remains open.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/970.Newer public claim (6 Oct 2026): OpenAI's internal-model release claims, for this problem, Jacobsthal: h(k) ≪ k²/(log log k)² — family 21 at github.com/openai/math, listed in its Lean catalogue. Bloom calls these claims, not endorsements; anything on this page that says the question is open was written before that release and has not re-checked it.
Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA048669
Original pageerdosproblems.com/970
Attempts
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FGKMT plus \(x=p_k\) rigorously strengthens the cited lower bound to \(H(k)\gg k(\log k)^2\log_3k/\log_2k\); an independent exhaustive checker gives exact \(H(k)\) for \(k\leq14\), but the quadratic u
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