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Claim. For every $\epsilon>0$ there are infinitely many $n$ such that the five consecutive integers $n,n+1,\dots,n+4$ are all $n^\epsilon$-smooth. The result is in section 2 of R. B. Eggleton and J. L. Selfridge, Consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 22 (1976), no. 1, 1–11, as the site's commentary credits it and as Balog and Wooley 1998 report it in their comparison (pp. 267–268): the smoothness bound is $\exp(c\log n/\log_3n)$ for runs of two or three and $\exp(c\log n/\sqrt{\log_3n})$ for runs of four or five, where $\log_3$ is the three-fold iterated logarithm, so each run is $n^\epsilon$-smooth once $n$ is large. Balog and Wooley also say that they correct a minor oversight in that section of the paper.

Covers. For every $\epsilon>0$ and every $k\le5$, infinitely many runs $n,\dots,n+4$ with each member $n^\epsilon$-smooth. This settles the question of Problem 369 for $k\le5$ (fix one such $n$; for every $N\ge n+4$ the run lies in $\{1,\dots,N\}$ and is $N^\epsilon$-smooth) and the site's first reading for $k\le5$, since each member $m\ge n$ is $m^\epsilon$-smooth. Not covered: $k\ge6$, and the site's second reading, a run inside $[N/2,N]$ for all large $N$, which infinitely many runs give only for infinitely many $N$; the full claims Yang 2026 and Bober, Fretwell, Martin and Wooley 2020 settle every $k$ in both readings, and the partial claim Balog and Wooley 1998 settles every $k$ of the wording and of the first reading.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: published in J. Austral. Math. Soc. Ser. A, a refereed journal (refereed). The site's curator credits the paper, in the problem's commentary, with the runs of five, and notes that this sits oddly with Erdős and Graham's remark that the problem was open even for $k=2$; the curator's PROVED (LEAN) label rests on the later results, not on this one, so no reviewed evidence is listed.

Dating. The page is dated by the issue month in the publisher's record, August 1976; the day is a placeholder.