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Claim. Fix $\theta\in(2/5,3/5)$ such that $(k,k+k^\theta)$ contains $\gg k^\theta/\log k$ primes for all large $k$ (Baker, Harman and Pintz give $\theta=21/40$). For all sufficiently large $k$ and every $n$ with $2k<n\le\exp(\log^2k/(20\log\log k))$, some prime $p\in(k,k+3k^\theta)$ divides $(n-k)\cdots(n-1)$. Since $3k^\theta<k$ for large $k$, this is $n_k>\exp(\log^2k/(20\log\log k))$ for all large $k$, with $n_k$ the quantity of Problem 451, and in particular $n_k>k^d$ for every fixed $d$ and all large $k$, which is Erdős's expectation of 1979. W. van Doorn and Q. Tang, Consecutive integers free of certain prime factors, arXiv:2606.19863v1 (18 June 2026), 5 pp.; Theorem 1.1, p. 1, paged as Theorem 1.1 of the library card. The proof imitates Konyagin's argument for the least prime factor of a binomial coefficient: if no prime of $(k,k+k^\theta)$ divides the product then $n/p$ lies within $k^{\theta-1}$ of an integer for every such $p$, and, after settling the smallest of four ranges of $n$ by exhibiting the factor directly, the paper bounds the number of $m$ in that interval with $\|n/m\|<k^{\theta-1}$ in the other three, the last two through a Konyagin-type inequality (Theorem 4.1, p. 3). Read depth: claims checked for Theorems 1.1 and 4.1; the proof was read for its structure and not checked step by step.

Submission note. Posted to the site's forum by Quanyu Tang on 26 April 2026:

After iterating GPT-5.5 Pro more than a dozen times, it produced a note claiming a good lower bound for $n_k$. More precisely, the draft claims to prove that there exists an absolute constant $c>0$ such that, for all sufficiently large $k$,

$$n_k>\exp\left(c\frac{(\log k)^2}{\log\log > k}\right).$$

In particular, this would imply Erdős's conjecture that $n_k>k^d$

for all constant $d$.

The draft has passed several AI-based checks, but I do not have time to manually verify every detail of the proof carefully. Therefore I am posting it here, and I would be very grateful for any comments, corrections, or independent checks.

The PDF is available here: pdfhere.

The tex source is available here: texhere.

History and provenance. The page is named by the arXiv paper of 18 June 2026, the source the site credits as [vDTa26]. The result's first posting is Tang's alone and states a weaker form: on 26 April 2026 Tang posted in the site's thread a note, produced by GPT 5.5 Pro after repeated prompting, claiming $n_k>\exp(c(\log k)^2/\log\log k)$ for an unspecified absolute $c>0$ and all large $k$, with a request for independent checks; van Doorn is an author only of the arXiv paper, which first states the theorem with the constant $1/20$. A reply of 27 April 2026 reported that a model-based check found no issue. The arXiv paper of 18 June 2026, announced in the thread on 19 June 2026, is the authors' human-written version of that note with the explicit constant $1/20$; its declaration of AI usage (p. 2) says that the text is entirely human-written, that the core idea of applying Konyagin's argument came from ChatGPT 5.5 Pro, whose write-up the authors keep in a public repository, and that Lean formalizations of all theorems were produced by Aristotle, Harmonic's automated prover; the site's commentary names the same model.

Covers. The lower bound, and with it Erdős's expectation that $n_k$ exceeds every fixed power of $k$, settled in the affirmative. Not covered: the estimate of $n_k$ itself, open between this bound and the elementary $n_k\le\prod_{k<p<2k}p=e^{(1+o(1))k}$; Erdős's expectation $n_k<e^{\varepsilon k}$ for every $\varepsilon>0$ and the heuristic $\log n_k\asymp k/\log k$ are unproved.

Standing. Claimed. The site's curator, Thomas Bloom, who is independent of the authors, took the theorem into the problem's commentary on 21 June 2026, stating the bound with an unspecified constant and crediting the argument to GPT 5.5 Pro and to Tang; the site labels the problem OPEN, so that commentary is not an acceptance of the claim and no evidence is listed. Not refereed: on 2026-09-18 the arXiv listing showed one version and no journal reference, and Crossref had no record. Not formalized in this corpus's sense: the Lean file in the repository of one author, linked above as the authors' own formalization (4,693 lines, pinned at its last change of 19 June 2026), declares one axiom, bhp, the Baker--Harman--Pintz count of primes in $(k,k+k^{21/40})$, and proves main_theorem, Theorem 1.1, relative to it with no sorry; it was neither built nor audited here, so the theorem holds relative to the declared axiom and the file is not acceptance evidence. The thread's second reply of 19 June 2026 is a congratulation, not a review. Nothing here is this project's own review.

Depends on. No page of this wiki.