Problem detail · source-aware
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Is it true that for every $\epsilon,\eta>0$ there exists a $k$ such that the density of $n$ for which $P(n(n+1)\cdots(n+k))>n^{1-\epsilon}$ is at least $1-\eta$, where $P(m)$ is the greatest prime divisor of $m$? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
The deduction from the Matomäki-Radziwiłł theorem on multiplicative functions was written by GPT-5.5 Pro; Tao and Sawin's forum discussion pinned down exactly what the known results do and do not give for this problem.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Discussed on the problem's forum, including by Tao, who delineated the remaining natural-density gap; no independent review of the note itself.
Correctness: unknown · statement fidelity: unaudited · peer review: none
As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.