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Claim. Let $k(n)$ be the largest $k$ for which some integer $m$ with $0\le m\le n$ has each of $m+1,\ldots,m+k$ divisible by a prime greater than $k$; the conclusions below hold equally when $m$ is required to be positive.

First, for all integers $K\ge64$ and $0\le m\le2^{-15}K^{28/11}$, the interval $(m,m+K]$ contains an integer $c(q-t)(q+t)$ whose three factors are positive integers at most $K$, hence an integer all of whose prime factors are at most $K$. Consequently

$$k(n)<\max\bigl(64,(2^{15}n)^{11/28}\bigr)\qquad(n\ge1),$$

and in Erdős's notation $n_k>2^{-15}k^{28/11}$ for every integer $k\ge64$, where $n_k$ is the least $n\ge0$ such that each of $n+1,\ldots,n+k$ has a prime factor greater than $k$.

Second, for every real $\delta$ with $0<\delta<1/20$ there is an integer $K_0(\delta)\ge64$ such that the same conclusion holds for all integers $K\ge K_0(\delta)$ and $0\le m\le K^{97/38-\delta}$. Consequently, for every $\varepsilon>0$ there is a constant $A_\varepsilon$ with

$$k(n)\le A_\varepsilon n^{38/97+\varepsilon}\qquad(n\ge1).$$

Covers. The estimate of $k(n)$ asked by Problem 962, which is open-ended: the upper exponent improves from the $2/5$ of Erdős and Sárközy's published bound $k(n)\ll n^{2/5}$ to $11/28\approx0.3929$ with explicit constants, and to every exponent above $38/97\approx0.3918$. No bound with exponent exactly $38/97$ and a fixed constant is claimed, and no lower bound. The displayed question, whether $\log k(n)\le(\log n)^{1/2+o(1)}$, stays open.

Credit. The construction of a smooth member $c(q-t)(q+t)$ of the block is Erdős and Sárközy's, from the proof of Theorem 2 of their 1979 paper; Plasma AI averages the multiplier $c$ over an interval. The first bound consumes Vaaler's inequality and the explicit third-derivative test as printed in Lemmas 7.1 and 8.6 of Granville and Ramaré (1996). The second applies Bourgain's exponent pair $(13/84+\eta,55/84+\eta)$, Theorem 6 of Decoupling, exponential sums and the Riemann zeta function (2017), consumed as a published input.

Standing. L251, L253 and L250 are tier 0: author-recorded by Plasma AI, with complete written proofs and no independent examination or Lean proof. Bourgain's decoupling proof is not reproduced or independently verified here. Catalog acceptance is pending.

Depends on. L253 (tier 0), L251 (tier 0) and L250 (tier 0), which covers the blocks with $m\le K^{5/2}/128$.