Erdős Problem #1201

The problem

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 $m$)?

Statement quoted from erdosproblems.com/1201, maintained by Thomas Bloom. Original source: [Er80,p.107]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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