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Claim. Let $n_k$ be the least integer $n$ such that each of $n+1,\ldots,n+k$ has a prime factor greater than $k$. Display (6) of Erdős's paper, printed p. 273 and paged at inequality (6), states that $n_k<k^{\log k/\log\log k}$ for $k>k_0$. The printed proof is four lines: the number of $k$-smooth integers below $n_k$ is at least $n_k/k$, since each block of $k$ consecutive integers below $n_k$ contains one, and comparing this with de Bruijn's asymptotic $U(k^\alpha,k)=(c_\alpha+o(1))k^\alpha$ for the count of $k$-smooth integers up to $k^\alpha$ bounds the exponent of $n_k$.

Some $m\le n$ works for $k$ in Problem 962 exactly when $n_k\le n$, so $k(n)=\max\{k:n_k\le n\}$. With $\log k=c\sqrt{\log n\log\log n}$ the exponent $(\log k)^2/\log\log k$ of (6) equals $(2c^2+o(1))\log n$, which is at most $\log n$ for large $n$ when $c<1/\sqrt2$; then $n_k\le n$ and $k(n)\ge k$. Hence $\log k(n)\ge(1/\sqrt2-o(1))\sqrt{\log n\log\log n}$. The site prints the weaker form $\log k(n)\gg\sqrt{\log n\log\log n}$.

Covers. The lower bound on $k(n)$ only. The estimate of $k(n)$ and the displayed question $\log k(n)\le(\log n)^{1/2+o(1)}$, the inverse of Erdős's conjecture (7), stay open.

Depends on. No page of this wiki; the proof rests on de Bruijn's smooth-number asymptotic (Indag. Math. 13 (1951), 50--60), which the paper cites.

Acceptance. Refereed: P. Erdős, Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), no. 3--4, 271--282. The site labels the problem OPEN, so its commentary crediting the argument is not reviewed evidence. The Crossref record gives the volume year without a day, so the day in the page name is a placeholder.