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Hung M. Bui, Slava Naprienko, Kyle Pratt and Alexandru Zaharescu answer the question in the negative (Binomial coefficients with divisors avoiding an interval, arXiv:2605.21221). Their Theorem 1.4 gives, for a large fixed $k_0$ and a small $\delta>0$, infinitely many coefficients $\binom nk$ with $k_0<k\leq\delta(\log\log n)^{1/2}$ and no divisor in

$$\Bigl(\frac{241\,n\log\log k}{\log k},\,n\Bigr],$$

and Corollary 1.6, as the third version prints it, records infinitely many $\binom nk$ with $k\asymp(\log\log n)^{1/2}$ and no divisor in $\bigl(500\,n\log\log\log\log n/\log\log\log n,\,n\bigr]$; the second version and the site's remark omit the factor $500$, without which the corollary does not follow from Theorem 1.4, since $241\log\log k/\log k$ is about $482\log\log\log\log n/\log\log\log n$ for such $k$. The lower end of that window is $o(n)$, so no constant $c>0$ puts a divisor of every $\binom nk$ in $(cn,n]$. In the other direction, Theorem 1.2 shows that for small $\varepsilon>0$, large $n$ and $\exp((\log n)^{2/3+\varepsilon})\leq k\leq n/2$ the coefficient has a divisor in $(n-n/(\log n)^{1/4},\,n]$, so the version Guy records, a divisor in $(cn,n]$ for every $c<1$ once $n$ is large, holds for $k$ in that range. The digest and result list are on the library card Bui, Naprienko, Pratt and Zaharescu 2026.

The result was posted in two steps. The first version (2026-05-20, by Bui, Pratt and Zaharescu) proved the negative answer under the Generalized Riemann Hypothesis. Naprienko's repository, the code link above, pinned at its last commit (2026-05-23), grew around it: a write-up of a residue-cover lemma, the Conjecture 1 Tao had posed in the site's discussion thread, on 2026-05-16, and a Lean 4 proof of that lemma on 2026-05-18, both before the first version; then on 2026-05-23, after it, the fixed-$B$ covering input that pairs with the divisor analysis of the paper's later sections and removes the need for the hypothesis. The second version (2026-06-30), by the four authors, is unconditional; the third version (2026-09-30) fixes small errors, the constant in Corollary 1.6 among them, adds Corollary 1.7 (for every fixed $c>0$ some $\binom nk$ with $1\leq k<n$ has no divisor in $(cn,n]$) and leaves the main theorems unchanged. Pratt announced the first and the unconditional versions in the site's discussion thread on 2026-05-21 and 2026-07-02. The paper's acknowledgments say that the main ideas in the proof of its main covering theorem (Theorem 5.1) were developed in interactive sessions between the authors and ChatGPT 5.5 Pro, that some documents and code in the repository were generated with AI assistance, and that ChatGPT was used for literature searches and for spotting misprints, with all text stated to be written by the authors (second version). The third version says instead that ChatGPT was used for literature searches, proofreading and checking earlier versions for possible mathematical errors, and drops the statement about the text.

The acceptance evidence is the site's curator, Thomas Bloom, who marks Problem 387 solved and credits the four authors' paper with the negative answer for every constant $c>0$ (page edited 2026-07-02). No journal publication is recorded so no refereed evidence is listed. The formal-conjectures file FormalConjectures/ErdosProblems/387.lean states the problem as erdos_387 : answer(False) ↔ … with sorry. The Lean in Naprienko's repository proves a weaker form of the paper's covering proposition (Proposition 5.5), with $B$ fixed rather than growing with $K$, which is not the version the proof uses, together with Tao's residue-cover conjecture; both rest on two project-local axioms, the prime number theorem in arithmetic progressions for a fixed modulus and a Siegel-Walfisz variant, and nothing has been built here, so the page lists no formalized evidence.