number-theory / Number Theory, Primes
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.
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Append-only history
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
Research memory
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.
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
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