The board / number theory / #945
Erdős Problem #945
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 \(F(x)\) be the maximal \(k\) such that there exist \(n+1, \dots, n+k \le x\) with \(τ(n+1), \dots, τ(n+k)\) all distinct, where \(τ(m)\) counts the divisors of \(m\).
Is it true that \(F(x) \leq (\log x)^{O(1)}\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/945.Prizenone
Topicsnumber theory, divisors
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA048892
Original pageerdosproblems.com/945
Attempts
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Dual exhaustive computation proves the exact first starts through \(k=13\), the full step function \(F(x)\) for integer \(x\leq20{,}000{,}000\), and no length-14 run there; an elementary reduction iso
PARTIAL