The board / number theory / #663
Erdős Problem #663
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(k \geq 2\), and let \(q(n,k)\) denote the least prime which does not divide \(\prod_{1 \leq i \leq k}(n+i)\). Is it true that, if \(k\) is fixed and \(n\) is sufficiently large, then \(q(n,k) < (1+o(1))\log n\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/663.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA391668
Original pageerdosproblems.com/663
Attempts
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exact CRT reduction plus a complete k=2 enumeration proves the sharp cutoff q(n,2)<=199 for every 1<=n<68979941211663467981891685180356798343731381481900118460455756739438, with q(N,2)=211 at equality
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