Erdős Problem #935

The problem

For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that

$$Q_2(n) = \prod_{\substack{p\\ k_p\geq 2}}p^{k_p}.$$

Is it true that, for every $\epsilon>0$ and $\ell\geq 1$, if $n$ is sufficiently large then

$$Q_2(n(n+1)\cdots(n+\ell))<n^{2+\epsilon}?$$

If $\ell\geq 2$ then is

$$\limsup_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^2}$$

infinite?

If $\ell\geq 2$ then is

$$\lim_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^{\ell+1}}=0?$$

Statement quoted from erdosproblems.com/935, maintained by Thomas Bloom. Original source: [Er76d]. 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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