The board / number theory / #710
Erdős Problem #710
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).
\(f(n)\): the least \(L\) such that the open interval \((n,n+L)\) contains distinct integers \(a_1,\ldots,a_n\) with \(k \mid a_k\) for all \(1 \leq k \leq n\). This is `Erdos711.HasDivisibleDistinctTuple n n L`. The set of such \(L\) is nonempty for every \(n\) (take \(a_k = k(n+1)\) and \(L = n^2+1\)), so `sInf` is the true minimum. For \(n = 0\) there is nothing to choose and \(f(0) = 0\).
Let \(f(n)\) be minimal such that in \((n,n+f(n))\) there exist distinct integers \(a_1,\ldots,a_n\) such that \(k\mid a_k\) for all \(1\leq k\leq n\). Obtain an asymptotic formula for \(f(n)\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/710.Prize₹2,000
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA390246
Original pageerdosproblems.com/710
Attempts
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Live page status is “OPEN - $78”, with SkyYang listed under “Currently working on this problem”; stopped to avoid colliding with a current worker.
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