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Claim. For every fixed $u>1$ there are infinitely many $n$ such that the $t(n)=\lfloor\log_4n/\log(3u)\rfloor$ consecutive integers $n+1,\dots,n+t(n)$ are all $n^{1/u}$-smooth, where $\log_4$ is the four-fold iterated logarithm (Theorem 1, p. 267). Since $t(n)\to\infty$, for every $\epsilon>0$ and $k\ge2$ there are infinitely many $m$ with $m+1,\dots,m+k$ all $m^\epsilon$-smooth. The source card is Balog and Wooley 1998.

Covers. The wording of Problem 369, which is trivially true, since $1,\dots,k$ are $n^\epsilon$-smooth once $n>k^{1/\epsilon}$, as the site notes; the theorem gives it with a nontrivial run: fix one of the infinitely many $m$; for every $n\ge m+k$ the run $m+1,\dots,m+k$ lies in $\{1,\dots,n\}$ and each member is $m^\epsilon$-smooth, hence $n^\epsilon$-smooth. The first of the two readings described on the problem page, that each member $x$ of the run be $x^\epsilon$-smooth, follows directly, since $m^\epsilon\le x^\epsilon$. The second, that the run lie in $[n/2,n]$, follows for infinitely many $n$ (take $n$ with $m+k\le n\le2m$). Not covered: the second reading for all large $n$, which needs the further construction of Yang 2026 or the theorem of Bober, Fretwell, Martin and Wooley 2020. The site's curator reports that Wooley described the paper's problem as slightly different, with a stronger uniformity that yields only infinitely many $n$.

Acceptance. Published in J. Austral. Math. Soc. Ser. A 64 (1998), no. 2, 266–276, a refereed journal, in the issue dated April 1998 in the publisher's record, which dates this page (refereed). The site's curator, Thomas F. Bloom, records in the problem's commentary (page last edited 2026-04-28) that the first strengthening follows from this result and that the second follows from it for infinitely many $n$ (reviewed). Eggleton and Selfridge [EgSe76] had earlier given, for every $\epsilon>0$, infinitely many runs of five consecutive integers $n,\dots,n+4$ each $n^\epsilon$-smooth, which the site notes sits oddly with Erdős and Graham's remark that the problem was open even for $k=2$; that result is the partial claim Eggleton and Selfridge 1976, and the paper's section 2 comparison (pp. 267–268) records its smoothness bounds and corrects a minor oversight in that result's source.

Depends on. Nothing in this wiki.