The board / number theory / #889
Erdős Problem #889
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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\(v(n,k)\) counts the prime factors of \(n+k\) which do not divide \(n+i\) for all \(0 \le i < k\).
\(v_0(n)\) is the supremum of \(v(n,k)\) for all \(k \ge 0\).
Let \(v(n,k)\) count the prime factors of \(n+k\) which do not divide \(n+i\) for \(0\leq i < k\). Is it true that \(v_0(n)=\max_{k\geq 0}v(n,k)\to \infty\) as \(n\to \infty\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/889.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/889
Attempts
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exact interval reduction and exhaustive verification give \(v_0(n)\ge3\) for every \(14{,}433{,}527\le n\le50{,}000{,}000\), with exactly 2,112 smaller-prefix exceptions and last \(14{,}433{,}526\); e
PARTIAL