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Claim. Theorem 2.1 of the paper: let $f(t)=\prod_{j=1}^{l}(a_jt^{k_j}-b_j)$ with integers $a_j\neq0$, $b_j$ and $k_j\geq1$. Then there is $c>0$, depending on the $k_j$, such that polynomials $g\in\mathbb{Z}[t]$ of arbitrarily large degree $d$ exist for which $f(g(t))$ factors as a product of polynomials of degree at most $cd/(\log\log d)^{1/l}$. Applied to $f(t)=t(t+1)\cdots(t+k-1)$, a product of $k$ linear factors, the theorem gives, for every $\epsilon>0$ and $k\geq2$, a constant $\theta<1$ such that for all large $n$ some $k$ consecutive integers in $[n-n^{\theta},n]$ are all $n^\epsilon$-smooth: for an integer $t$ the $k$ consecutive integers $g(t),\dots,g(t)+k-1$ have every prime factor bounded by a polynomial value of degree at most $cd/(\log\log d)^{1/k}$ in $t$, hence by $n^{\epsilon}$ once $d$ is large, and consecutive values of $g$ are $O(n^{1-1/d})$ apart, so every large $n$ has such a run in $[n-O(n^{1-1/d}),n]$, inside $[n-n^{\theta},n]$ for any $\theta\in(1-1/d,1)$. This deduction was pointed out by Wooley to the site's curator, Thomas F. Bloom, who wrote it out on the forum on 2026-03-27 and records its conclusion in the problem's commentary. The run lies in $[n/2,n]$, so the result settles the question of Problem 369 as written, the formal-conjectures statement erdos_369, and both strengthenings the site proposes: the second directly, and the first because a run of $n^{\epsilon/2}$-smooth integers above $n/2$ is $m^{\epsilon}$-smooth for each of its members $m$. It is stronger than the result of Yang 2026, whose run lies in $(3n/4,n)$, and than that of Balog and Wooley 1998, which gives infinitely many $n$. The source card is Bober, Fretwell, Martin and Wooley 2020.

Acceptance. Published in J. Aust. Math. Soc. 108 (2020), no. 2, 245–261, a refereed journal, online 2019-02-01 in the publisher's record (refereed); the arXiv posting of 2017-10-05 dates this page. The site's curator labels the problem PROVED (LEAN) and credits the paper's main result, in the problem's commentary (page last edited 2026-04-28), with the stronger statement above (reviewed). The Lean proofs linked by formal-conjectures formalize Yang's construction, not this theorem, so formalized is not listed.

Depends on. Nothing in this wiki.